diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzzjlif" "b/data_all_eng_slimpj/shuffled/split2/finalzzjlif" new file mode 100644--- /dev/null +++ "b/data_all_eng_slimpj/shuffled/split2/finalzzjlif" @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\\label{sec:intro}\n\t\tIn its original form, the quantum Zeno effect is defined for closed finite quantum systems. \\citeauthor{Misra.1977} predicted that \"an unstable particle which is continuously observed to see whether it decays will never be found to decay!\" \\cite[Abst.]{Misra.1977}. In a more general setup, frequent measurements enable a change in the time evolution and convergence to the so called \\textit{Zeno dynamics}. Experimentally, the Zeno effect is verified for instance in \\cite{Itano.1990, Fischer.2001}. In addition to its theoretical value, the quantum Zeno effect is used in error correction schemes to suppress decoherence in open quantum systems \\cites{Hosten.2006}{Franson.2004}{Beige.2000}{Barankai.2018}{Luchnikov.2017}. The idea is to frequently measure the quantum state and thereby force the evolution to remain within the code space. With an appropriate measurement, one can even decouple the system from its environment \\cites{Facchi.2004}{Burgarth.2020} and show that appropriately encoded states can be protected from decoherence with arbitrary accuracy \\cites{Dominy.2013}{Erez.2004}. Moreover, the quantum Zeno effect has been used in commercial atomic magnetometers \\cite{Kominis.2009}.\n\t\t\n\t\tIntroduced by \\citeauthor{Beskow.1967} in 1967 and later named by \\citeauthor{Misra.1977} after the greek philosopher Zeno of Elea, the quantum Zeno effect in its simplest form can be stated as follows: given a projective measurement $P$ and a unitary time-evolution generated by a Hamiltonian $H$ acting on a finite dimensional Hilbert space $\\cH$ \\cite{Misra.1977}: For $n\\to\\infty$\n\t\t\\begin{equation}\\label{eq:zeno-misra-sudarshan}\n \t\t(Pe^{\\frac{it}{n}H})^n\\longrightarrow e^{it\\,PHP}\n\t\t\\end{equation}\n\t Since the seminal works \\cites{Beskow.1967}{Misra.1977}, the result was extended in many different directions (overviews can be found in \\cites{Facchi.2008}{Schmidt.2004}{Itano.1990}). Recently, the convergence in \\Cref{eq:zeno-misra-sudarshan} was proven in the strong topology for unbounded Hamiltonian under the weak assumption that $PHP$ is the generator of a $C_0$-semigroup \\cite{Exner.2021}. Earlier approaches used the so called \\textit{asymptotic Zeno condition} \\cites{Schmidt.2004}{Exner.1989}{Misra.1977}, which assumes $(\\1-P)e^{itH}P$ and $Pe^{itH}(\\1-P)$ to be Lipschitz continuous at $t=0$. This condition is natural in the sense that it is related to the boundedness of the first moment of the Hamiltonian in the initial state and is efficiently verifiable in practice. With the works \\cites{Burgarth.2020}{Mobus.2019}{Barankai.2018}, the quantum Zeno effect was generalized to open and infinite dimensional quantum systems equipped with general quantum operations and uniformly continuous time evolutions. Note that in open quantum systems, we are dealing with operators acting on the Banach space $\\mathcal{T}_1(\\cH)$ of trace-class operators. More recently, \\citeauthor{Becker.2021} generalized\n\t\tthe Zeno effect further and interpreted the Zeno sequence as a product formula consisting of a contraction $M$ (quantum operation) and a $C_0$-contraction semigroup (quantum time evolution) on an abstract Banach space. Under a condition of \\textit{uniform power convergence} of the power sequence $\\{M^k\\}_{k\\in\\mathbb{N}}$ towards a projection $P$ and boundedness of $M\\cL $ and $\\cL M$, they proved a quantitative bound on the convergence rate \\cite{Becker.2021}:\n\t\t\\begin{align}\\label{eq:Becker}\n\t\t\t\\|(Me^{\\frac{t}{n}\\cL})^nx-e^{tP\\cL P}Px\\|=\\cO(n^{-\\frac{1}{3}}(\\|x\\|+\\|\\cL x\\|))\\,,\n\t\t\\end{align} \n\t\tfor $n\\rightarrow\\infty$ and all $x\\in\\cD(\\cL)$. However, the optimality of \\eqref{eq:Becker} was left open.\\footnote{Note that we found an inconsistency in the proof of \\cite[Lem.~2.1]{Zagrebnov.2017} (see \\cite{Zagrebnov.2022}), which slightly reduces the convergence rate found in \\Cref{eq:Becker} (more details are given in \\Cref{sec:alternative-chernoff-lemma-trotter-product-formula}).}\n\t\t\n\t\t\\subsubsection*{Main contributions:}\n\t\t\tIn this paper, we achieve the optimal convergence rate $\\mathcal{O}(n^{-1})$ of the Zeno sequence consistent with the finite-dimensional case \\cite{Burgarth.2020} by providing an explicit bound which recently attracted interest in finite closed quantum systems \\cite[Thm.~1]{Burgarth.2021}. Moreover, we generalize the results of \\cite{Becker.2021} in two complementary directions:\n\t\t\t\n\t\t\tIn \\Cref{thm:spectral-gap} below, we assume a special case of the \\textit{uniform power convergence} assumption on $M$, that is $\\|M^n-P\\|\\leq\\delta^n$ for some $\\delta\\in(0,1)$, and weaken the assumption on the semigroup to the \\textit{uniform asymptotic Zeno condition} inherited from the unitary setting of \\cite{Schmidt.2004}: for $t\\rightarrow0$\n\t\t\t\\begin{equation*}\n\t\t\t\t\\norm{(\\1-P) e^{t\\cL}P}_\\infty=\\cO(t)\\quad\\text{and}\\quad \\norm{Pe^{t\\cL}(\\1-P)}_\\infty=\\cO(t).\n\t\t\t\\end{equation*}\n\t\t\tTherefore, we prove the convergence of a non-trivial Zeno sequence in open quantum systems to a Zeno dynamics described by a possibly unbounded generator.\\\\\n\t\t\tSecond, \\Cref{thm:spectral-gap-uniform} is stated under slightly weaker assumptions as Theorem 3 in \\cite{Becker.2021} and improves the result to the optimal convergence rate and to the uniform topology.\n\t\t\n\t\t\tIn order to achieve these results, we prove a modified Chernoff $\\sqrt{n}$-Lemma in \\Cref{lem:improved-chernoff}, find a quantitative convergence rate for $\\operatorname{exp}\\big({nP(e^{\\frac{1}{n}t\\cL}-\\1)P}\\big)P-\\operatorname{exp}(tP\\cL P)P$ as $n \\rightarrow\\infty$, where $P\\cL P$ is possibly unbounded, and prove the upper semicontinuity of parts of the spectrum of $Me^{t\\cL}$ under tight assumptions.\n\t\t\n\t\t\\subsubsection*{Organization of the paper:}\n\t\t\tIn \\Cref{sec:prelim}, we provide a short recap on bounded and unbounded operator theory. We expose our main results in \\Cref{sec:assumptions-results}. \\Cref{sec:alternative-chernoff-lemma-trotter-product-formula} deals with the modified Chernoff $\\sqrt{n}$-Lemma and some of its implications as regards to Trotter-Kato's product formula. Then, we prove our main theorems under the weakest possible assumptions on the $C_0$-semigroup in \\Cref{sec:unbounded-generator-zeno-subspace}, and under the weakest possible assumptions on $M$ in \\Cref{sec:finitely-many-eigenvalues}. Our results are illustrated by three large classes of examples in finite and infinite dimensional quantum systems in \\Cref{sec:applications}. Finally, we discuss some remaining open questions in \\Cref{sec:discussion}.\n\t\t\n\t\t\n\t\\section{Preliminaries}\\label{sec:prelim}\n\t\tLet $(\\cX,\\|\\cdot\\|)$ be a Banach space and $(\\cB(\\cX),\\|\\cdot\\|_{\\infty})$ be the associated space of bounded linear operators over $\\C$ equipped with the operator norm, i.e.~$\\|C\\|_{\\infty}\\coloneqq\\sup_{x\\in\\cX\\backslash\\{0\\}}\\frac{\\|Cx\\|}{\\|x\\|}$, and the identity $\\1\\in\\cB(\\cX)$. By a slight abuse of notation, we extend all densely defined and bounded operators by the \\textit{bounded linear extension theorem} to bounded operators on $\\cX$ \\cite[Thm.~2.7-11]{Kreyszig.1989}. \n\t\tA sequence $(C_k)_{k\\in\\N}\\subset\\cB(\\cX)$ converges uniformly to $C\\in\\cB(\\cX)$ if $\\lim_{k\\rightarrow\\infty}\\|C_k-C\\|_\\infty=0$ and strongly if $\\lim_{k\\rightarrow\\infty}\\|C_kx-Cx\\|=0$ for all $x\\in\\cX$. The integral over bounded vector-valued maps, e.g.~$[a,b]\\rightarrow\\cX$ or $[a,b]\\rightarrow \\cB(\\cX)$ with $a0$ depending on $t$ and $b$, but independent of $n$ .\n\t\t\\end{manualthm}\n\t\t\\begin{rmk*}\n\t\t\tNote that the assumption on $M$ is a special case of the so-called \\textit{uniform power convergence} assumption (q.v.~\\Caref{unifpower-prev}) and the assumption (\\ref{eq:thm1-asympzeno-prev}) on the $C_0$-semigroup is a generalization of the uniform \\textit{asymptotic Zeno condition} which implies the convergence in the case of a unitary evolution frequently measured by a projective measurement \\cite[Sec.~3.1]{Schmidt.2004}. Note that in that specific case, \\cite{Exner.2021} recently managed to remove the asymptotic Zeno condition. Moreover, the assumption that $(P\\cL P,\\cD(\\cL P))$ is a generator can be relaxed to the assumption that $P\\cL P$ is closeable and its closure defines a generator (q.v.~remark after \\Caref{lem:proofthm1-term3}). The famous \\textit{Generation Theorem} by Hille and Yosida provides a sufficient condition under which $\\overline{P\\cL P}$ is a generator \\cite[Thm.~3.5-3.8]{Engel.2000}.\n\t\t\\end{rmk*}\n\t The following example confirms the optimality of the achieved convergence rate.\n\t\t\\begin{ex}\n\t\t\tLet $\\left\\{\\ket{1},\\ket{2},\\ket{3}\\right\\}$ be an orthonormal basis of $\\R^3$ and $\\delta\\in(0,1)$. We define, \n\t\t\t\\begin{equation*}\n\t\t\t\t\\cL\\coloneqq \\ket{1}\\bra{2}\\quad\\text{and}\\quad M\\coloneqq\\ket{1}\\bra{1}+\\delta\\ket{3}\\bra{3}.\n\t\t\t\\end{equation*}\n\t\t\tThen, $P=\\ket{1}\\bra{1}$, $(\\1-P) M=\\delta\\ket{3}\\bra{3}$, $M\\cL=\\cL M$, and $\\cL^2=0=\\cL P$, $\\|M^n-P\\|_\\infty\\leq\\delta^n$. Using these properties, $Me^{\\frac{t}{n}\\cL}=M +\\frac{t}{n}\\cL$ and for $t\\in[0,\\infty)$\n\t\t\t\\begin{align*}\n\t\t\t \\left(Me^{\\frac{t}{n}\\cL}\\right)^n&=\\left(M+\\frac{t}{n}\\cL\\right)^n\\\\\n\t\t\t &=\\delta^n\\ket{3}\\bra{3}+\\ket{1}\\bra{1}+\\frac{t}{n}\\ket{1}\\bra{2}\\\\\n\t\t\t &=\\left((\\1-P)M\\right)^ne^{t\\cL}+Pe^{tP\\cL P}+\\frac{t}{n}P\\cL.\n\t\t\t\\end{align*}\n\t\t\tTherefore, \n\t\t\t\\begin{equation*}\n\t\t\t\t\\norm{\\left(Me^\\frac{t}{n}\\cL\\right)^n-Pe^{tP\\cL P}}_\\infty=\\max\\{\\tfrac{t}{n},\\delta^n\\},\n\t\t\t\\end{equation*}\n\t\t\twhich shows the optimality of our convergence rate in \\Cref{thm:spectral-gap}.\n\t\t\\end{ex}\n Beyond the proven asymptotics, we find explicit error bounds in \\Cref{lem:proofthm1-term1}, \\ref{lem:proofthm1-term2}, and \\ref{lem:proofthm1-term3}, which simplify if $\\cL$ is bounded to the following explicit convergence bound depending on the generator $\\cL$, the projection $P$, the spectral gap $\\delta$, and the time $t$:\n\t\t\\begin{prop}\\label{cor:explicit-bound-thm1}\n\t\t\tLet $\\cL\\in\\cB(\\cX)$ be the generator of a contractive uniformly continuous semigroup and $M\\in\\cB(\\cX)$ a contraction satisfying\n\t\t\t\\begin{equation*}\n\t\t\t\t\\norm{M^n-P}_\\infty\\leq\\delta^n\n\t\t\t\\end{equation*}\n\t\t\tfor a projection $P\\in\\cB(\\cX)$, $\\delta\\in(0,1)$, and all $n\\in\\N$. Then, for all $t\\geq0$ and $n\\in\\N$,\n\t\t\t\\begin{align*}\n\t\t \\norm{\\left(Me^{\\frac{t}{n}\\cL}\\right)^n-e^{tP\\cL P}P}_\\infty\\leq c_{p}\\frac{t\\|\\cL\\|_\\infty}{n} &+\\left(\\frac{c_{p}+(1+e^{\\tilde{b}})(1+c_{p}^2)}{2}\\right)\\frac{t^2\\|\\cL\\|_\\infty^2}{n}\\\\\n\t\t &+\\delta^n+\\frac{2\\delta}{1-\\delta}\\frac{e^{3t\\|\\cL\\|_\\infty c_{p}}}{n}\n\t \t\\end{align*}\n\t\t\twhere $c_{p}\\coloneqq\\|\\1-P\\|_\\infty$ and $e^{\\tilde{b}}=\\sup_{s\\in[0,t]}\\|e^{sP\\cL P}\\|_\\infty$.\n\t\t\\end{prop}\n\t\tNote that the above proposition can be easily extended to the case of an unbounded generator with the assumption that $\\cL M$ and $M\\cL$ are densely defined and bounded. Another advantage of our setup is the freedom it provides for choosing the Banach space $\\cX$, which allows us to treat open quantum systems ($\\cX=\\cT(\\cH)$ the trace class operators over a Hilbert space) and closed quantum systems ($\\cX=\\cH$ a Hilbert space) on the same footing. In the case of finite dimensional closed quantum systems, \\Cref{cor:explicit-bound-thm1} reduces to the following bound, which was independently proven in \\cite[Thm.~1]{Burgarth.2021} (up to a change of the numerical constant in the quadratic term from $\\tfrac{5}{2}$ to $2$):\n\t\n\t\t\\begin{cor}\\label{cor:explicit-bound-thm1-closed-sys}\n\t\t\tLet $\\cH$ be a Hilbert space, $H\\in\\cB(\\cH)$ be a hermitian operator, and $P\\in\\cB(\\cH)$ a hermitian projection. Then,\n\t\t\t\\begin{equation*}\n\t\t\t\t\\norm{\\left(Pe^{-i\\frac{t}{n}H}\\right)^n-e^{-itPH P}P}_\\infty\\leq\\frac{1}{n}\\left(t\\norm{H}_\\infty+\\frac{5}{2}t^2\\norm{H}_\\infty^2\\right)\n\t\t\t\\end{equation*}\n\t\t\\end{cor}\n\t\tTo achieve the bound above, one inserts $\\delta=0$ and $\\cL=iH$ in \\Cref{cor:explicit-bound-thm1}. Note that $PHP$ is hermitian and $\\|e^{siPHP}\\|_\\infty=1$ for all $s\\geq0$. \n\t\t\n\t\tNext, we consider convergence rates under a slight weakening of the condition on the map $M$:\n\n\t\t\\begin{cor}\\label{cor:spectral-gap}\n\t\t\tLet $(\\cL,\\cD(\\cL))$ be the generator of a $C_0$-contraction semigroup on $\\cX$ and $M\\in\\cB(\\cX)$ a contraction satisfying \n\t\t\t\\begin{equation*}\n\t\t\t\t\\norm{M^n-P}_\\infty\\leq\\tilde{c}\\,\\delta^n\n\t\t\t\\end{equation*}\n\t\t\tfor some projection $P$, $\\delta\\in(0,1)$, $\\tilde{c}\\ge 0$ and all $n\\in\\N$. Moreover, assume there is $b\\geq0$ so that \n\t\t\t\\begin{equation*}\n\t\t\t\t\\norm{Pe^{t\\cL}P^\\perp}_\\infty\\leq tb\\quad\\text{and}\\quad\\norm{P^\\perp e^{t\\cL}P}_\\infty\\leq tb.\n\t\t\t\\end{equation*}\n\t\t\tIf $(P\\cL P,\\cD(\\cL P))$ is the generator of a $C_0$-semigroup, then there exists a constant $c>0$ and $n_0\\in\\N$ so that $\\tilde{c}\\delta^{n_0}\\eqqcolon\\tilde{\\delta}<1$ and for all $x\\in\\cD((\\cL P)^2)$\n\t\t\t\\begin{equation*}\n\t\t\t\t\\norm{\\left(M^{n_0}e^{\\frac{t}{n}\\cL}\\right)^nx-e^{tP\\cL P}Px}\\leq\\frac{c}{n}\\left(\\norm{x}+\\norm{\\cL P x}+\\norm{(\\cL P)^2x}\\right)+\\tilde{\\delta}^n\\norm{x}.\n\t\t\t\\end{equation*}\n\t\t\\end{cor}\n\t\tThe above \\namecref{cor:spectral-gap} follows by the choice $n_0$ such that $\\tilde{c}\\delta^{n_0}<1$ and applying \\Cref{thm:spectral-gap-prev} to $\\tilde{M}\\coloneqq M^{n_0}$. A more physically motivated result treating the same generalization as the \\namecref{cor:spectral-gap} above is provided in the next result:\n\t\t\n\t\t\\begin{manualprop}{II}[stated as~\\Caref{prop:spectral-gap-uniform-norm-power-convergence} in main text]\\label{prop:spectral-gap-uniform-norm-power-convergence-prev}\n Let $(\\cL,\\cD(\\cL))$ be the generator of a $C_0$-contraction semigroup on $\\cX$ and $M\\in\\cB(\\cX)$ a contraction such that\n \\begin{equation}\\label{unifpowerc-prev}\n\t\t\t\t\\norm{M^n-P}_\\infty\\leq\\tilde{c}\\,\\delta^n\n\t\t\t\\end{equation}\n\t\t\tfor some projection $P$, $\\delta\\in(0,1)$ and $\\tilde{c}\\ge 0$. Moreover, we assume that there is $b\\geq0$ so that \n\t\t\t\\begin{equation}\\label{key-prev}\n\t\t\t\t\\norm{Pe^{t\\cL}P^\\perp}_\\infty\\leq tb,\\quad\\norm{M^\\perp e^{t\\cL}-M^\\perp}_\\infty\\leq tb,\\quad\\text{and}\\quad\\norm{P^\\perp e^{t\\cL}P}_\\infty\\leq tb\n\t\t\t\\end{equation}\n\t\t\twhere $M^\\perp=(\\1-P)M$. If $(P\\cL P,\\cD(\\cL P))$ generates a $C_0$-semigroup, then there is an $\\epsilon>0$ such that for all $t\\geq0$, $n\\in\\N$ satisfying $t\\in[0,n\\epsilon]$, $\\tilde{\\delta}\\in(\\delta,1)$, and $x\\in\\cD((\\cL P)^2)$ \n\t\t\t\\begin{equation*}\n\t\t\t\t\\norm{\\left(Me^{\\frac{t}{n}\\cL}\\right)^nx-e^{tP\\cL P}Px}\\leq\\frac{c_1(t,b,\\tilde{\\delta}-\\delta)}{n}\\left(\\norm{x}+\\norm{\\cL P x}+\\norm{(\\cL P)^2x}\\right)+c_2(\\tilde{c},\\tilde{\\delta}-\\delta)\\tilde{\\delta}^n\\norm{x}\\,,\n\t\t\t\\end{equation*}\n\t\t for some constants $c_1,c_2\\ge 0$ depending on $t$, $b$, the difference $\\tilde{\\delta}-\\delta$, and $\\tilde{c}$.\n\t\t\\end{manualprop}\n\t\t\n\t\tAs in \\Cref{cor:explicit-bound-thm1}, we also get a more explicit bound in the case of bounded generators in the following proposition:\n\t\t\\begin{prop}\\label{cor:explicit-bound-prop}\n\t\t\tLet $\\cL\\in\\cB(\\cX)$ be the generator of a contractive uniformly continuous semigroup and $M\\in\\cB(\\cX)$ a contraction satisfying\n\t\t\t\\begin{equation*}\n\t\t\t\t\\norm{M^n-P}_\\infty\\leq \\tilde{c}\\delta^n\n\t\t\t\\end{equation*}\n\t\t\tfor a projection $P\\in\\cB(\\cX)$, $\\delta\\in(0,1)$, $\\tilde{c}>1$, and all $n\\in\\N$. Then there is $\\epsilon>0$ such that for all $t\\geq0$, $n\\in\\N$ satisfying $t\\in[0,n\\epsilon]$, and $\\tilde{\\delta}\\in(\\delta,1)$\n\t\t\t\\begin{align*}\n\t \\norm{\\left(Me^{\\frac{t}{n}\\cL}\\right)^n-e^{tP\\cL P}P}_\\infty\\leq\\frac{tc_{p}\\|\\cL\\|_\\infty}{n}&+\\frac{c_{p}+(1+e^{\\tilde{b}})(1+c_{p}^2)}{2}\\frac{t^2\\|\\cL\\|_\\infty^2}{n}\\\\\n\t &+\\frac{2\\tilde{c}}{\\tilde{\\delta}-\\delta}\\tilde{\\delta}^n+\\frac{2\\tilde{\\delta}}{1-\\tilde{\\delta}}\\frac{e^{\\frac{6tc_{p}\\tilde{c}\\|\\cL\\|_\\infty}{\\tilde{\\delta}-\\delta}}}{n}\n\t \\end{align*}\n\t where $c_{p}\\coloneqq\\|\\1-P\\|_\\infty$ and $e^{\\tilde{b}}=\\sup_{s\\in[0,t]}\\|e^{sP\\cL P}\\|_\\infty$.\n\t\t\\end{prop}\n\t\tFinally, we extend the assumption on $M$ (q.v.~\\Caref{unifpower1-prev}, (\\ref{unifpowerc-prev})) to the \\textit{uniform power convergence} introduced in \\cite{Becker.2021}. Let $\\{P_j,\\lambda_j\\}_{j=1}^J$ be a set of projections satisfying $P_jP_k=1_{j=k} P_j$ and associated eigenvalues on the unit circle $\\partial\\D_1$. Then, $M$ is called uniformly power convergent with rate $\\delta\\in(0,1)$ if $M^n-\\sum_{j=1}^{J}\\lambda^n_jP_j=\\cO(\\delta^n)$ uniformly for $n\\rightarrow\\infty$. To prove our result in this case, we also need to assume that $M\\cL$ and $\\cL P_\\Sigma$ with $P_\\Sigma\\coloneqq\\sum_{j=1}^{J}P_j$ are densely defined and bounded (cf.~\\cite{Becker.2021}) and $P_j$ is a contraction for all $j\\in\\{1,...,J,\\Sigma\\}$:\n\t\t\n \\begin{manualthm}{III}[stated as~\\Caref{thm:spectral-gap-uniform} in main text]\\label{thm:spectral-gap-uniform-prev}\n\t\t Let $(\\cL,\\cD(\\cL))$ be the generator of a $C_0$-contraction semigroup on $\\cX$ and $M\\in\\cB(\\cX)$ a contraction satisfying the following uniform power convergence: there is $\\tilde{c}>0$ so that\n\t\t\t\\begin{equation}\\label{unifpower-prev}\n\t\t\t \\norm{M^n-\\sum_{j=1}^{J}\\lambda^n_jP_j}_\\infty\\leq\\tilde{c}\\,\\delta^n\n\t\t \\end{equation}\n\t\t for a set of projections $\\{P_j\\}_{j=1}^J$ satisfying $P_jP_k=1_{j=k}P_j$, eigenvalues $\\{\\lambda_j\\}_{j=1}^J\\subset\\partial\\D_1$, and a rate $\\delta\\in(0,1)$. For $P_\\Sigma\\coloneqq\\sum_{j=1}^{J}P_j$, we assume that $M\\cL$ and $\\cL P_\\Sigma$ are densely defined and bounded by $b\\geq0$ and $\\|P_j\\|_\\infty=1$ for all $j\\in\\{1,...,J,\\Sigma\\}$. Then, there is an $\\epsilon>0$ such that for all $n\\in\\N$, $t\\geq0$ satisfying $t\\in[0,n\\epsilon]$, and $\\tilde{\\delta}\\in(\\delta,1)$\n\t\t\t\\begin{equation*}\n\t\t\t\t\\norm{\\left(Me^{\\frac{t}{n}\\cL}\\right)^n-\\sum_{j=1}^{J}\\lambda_j^ne^{tP_j\\cL P_j}P_j}_\\infty\\leq\\frac{c_1 }{n}+c_2\\tilde{\\delta}^n\\,,\n\t\t\t\\end{equation*}\n\t\t\tfor some constants $c_1,c_2\\ge 0$ depending on all involved parameters except from $n$.\n \\end{manualthm}\n \n\t\tIn comparison to Theorem 3 in \\cite{Becker.2021}, \\Cref{thm:spectral-gap-uniform} achieves the optimal convergence rate and is formulated in the uniform topology under slightly weaker assumptions on the generator.\n\t\t\\begin{rmk*}\n\t\t\tA natural way to weaken the above assumption is to assume that the power converges is in the strong topology (cf.~\\cite[Thm.~2]{Becker.2021}).\n\t\t\\end{rmk*}\n\t\t\n\t\t\n\t\\section{Chernoff \\texorpdfstring{$\\sqrt{n}$-}{}Lemma and Trotter-Kato's Product Formula}\\label{sec:alternative-chernoff-lemma-trotter-product-formula}\n\t\tIn previous works \\cite{Mobus.2019,Becker.2021}, Chernoff's $\\sqrt{n}$-Lemma \\cite[Lem.~2]{Chernoff.1968}, which we restate here, is used as a proof technique to approximate the Zeno product by a semigroup (q.v.~\\Caref{eq:proofthm1-term2}).\n\t\t\\begin{lem}[Chernoff \\texorpdfstring{$\\sqrt{n}$-}{square root }Lemma]\\label{lem:chernoff}\n\t\t\tLet $C\\in\\cB(\\cX)$ be a contraction. Then, $(e^{t(C-\\1)})_{t\\geq0}$ is a uniformly continuous contraction semigroup and for all $x\\in\\cX$\n\t\t\t\\begin{equation*}\n\t\t\t\t\\norm{C^nx-e^{n(C-\\1)}x}\\leq\\sqrt{n}\\norm{(C-\\1)x}.\n\t\t\t\\end{equation*}\n\t\t\\end{lem}\n\t\t\\begin{rmk*}\n\t\t\tIn Lemma 2.1 in \\cite{Zagrebnov.2017}, the dependence on $n$ is improved to $n^\\frac{1}{3}$. This is crucial in the proof of the convergence rate in \\cite[Lem.~5.4-5.5]{Becker.2021}. Unfortunately, we found an inconsistency in the proof of \\cite[Lem.~2.1]{Zagrebnov.2017}, i.e.~Inequality 2.3 is not justified. An update and more Chernoff bounds can be found in \\cite{Zagrebnov.2022}. Following the proof by \\citeauthor{Becker.2021}, one can achieve a convergence rate of order $\\tfrac{1}{\\sqrt{n}}$ in the bounded generator case \\cites[Thm.~1]{Becker.2021} and of order $\\tfrac{1}{\\sqrt[4]{n}}$ in the unbounded generator case \\cites[Thm.~3]{Becker.2021}.\n\t\t\\end{rmk*}\n\t\tIn the case of the quantum Zeno effect (see \\Caref{lem:proofthm1-term2} and \\ref{lem:proofthm2-term2}) for bounded generators, the contraction $C$ is a vector-valued map $t\\mapsto C(t)$ on $\\cX$ satisfying $\\|C(\\tfrac{1}{n})-\\1\\|_\\infty=\\cO(n^{-1})$. By Chernoff's $\\sqrt{n}$-Lemma \n\t\t\\begin{equation*}\n\t\t\t\\norm{C^n(\\tfrac{1}{n})-e^{n(C(\\tfrac{1}{n})-\\1)}}_\\infty\\leq\\frac{1}{\\sqrt{n}}.\n\t\t\\end{equation*}\n\t\tHere, we chose the bounded generator case for sake of simplicity. Nevertheless, the argument can be extended to unbounded generator as well (see \\Caref{lem:proofthm1-term2}, \\ref{lem:proofthm2-term2}, and \\ref{lem:approx-improved-chernoff}). Next, we prove a modified bound, which allows us to achieve the optimal rate in the quantum Zeno effect.\n\t\t\\begin{lem}[Modified Chernoff Lemma]\\label{lem:improved-chernoff}\n\t\t\tLet $C\\in\\cB(\\cX)$ be a contraction and $n\\in\\N$. Then, $(e^{t(C-\\1)})_{t\\geq0}$ is a contraction semigroup and for all $x\\in\\mathcal{X}$\n\t\t\t\\begin{equation*}\n\t\t\t\t\\norm{\\left(C^n-e^{n(C-\\1)}\\right)x}\\leq\\frac{n}{2}\\norm{(C-\\1)^2x}.\n\t\t\t\\end{equation*}\n\t\t\\end{lem}\n\t\t\\begin{rmk*}\n\t\t\tAt first glance, this seems to be worse than the original Chernoff $\\sqrt{n}$-Lemmas in \\cites{Chernoff.1968}. However, if $C$ is a vector-valued map satisfying $\\|C(\\tfrac{1}{n})-\\1\\|_\\infty=\\cO(n^{-1})$, then the modified Chernoff lemma gives\n\t\t\t\\begin{equation*}\n\t\t\t\t\\norm{C(\\tfrac{1}{n})^n-e^{n(C(\\text{\\tiny{$\\tfrac{1}{n}$}})-\\1)}}_\\infty\\leq\\frac{n}{2}\\norm{C(\\tfrac{1}{n})-\\1}_{\\infty}^2=\\cO(n^{-1})\n\t\t\t\\end{equation*}\n\t\t\twhich is the key idea to prove the optimal convergence rate of the quantum Zeno effect for bounded generators and contractions $M$ satisfying the uniform power convergence (q.v.~\\Caref{lem:proofthm1-term2}).\n\t\t\\end{rmk*}\n\t\t\\begin{proof}[Proof of \\Cref{lem:improved-chernoff}]\n\t\t\tSimilar to Chernoff's proof \\cite[Lem.~2]{Chernoff.1968}, $(e^{t(C-\\1)})_{t\\geq0}$ is a contraction semigroup. We define $C_t\\coloneqq (1-t)\\1+tC=\\1+t(C-\\1)$ for $t\\in[0,1]$, which itself is a contraction as a convex combination of contractions, and we use the fundamental theorem of calculus so that\n\t\t\t\\begin{align*}\n\t\t\t\t\\norm{\\left(C^n-e^{n(C-\\1)}\\right)x}&\\leq\\int_{0}^{1}\\norm{\\frac{\\partial}{\\partial t}(C_t^ne^{(1-t)n(C-\\1)})x}dt\\\\\n\t\t\t\t&\\leq n\\int_{0}^{1}\\norm{C_t^{n-1}e^{(1-t)n(C-\\1)}}_\\infty\\norm{(\\1-C)(\\1-C_t)x}dt\\\\\n\t\t\t\t&\\leq\\frac{n}{2}\\norm{(C-\\1)^2x},\n\t\t\t\\end{align*}\n\t\t\twhich proves the \\namecref{lem:improved-chernoff}.\n\t\t\\end{proof}\n\t\tIn \\cite[p.~241]{Chernoff.1968}, Chernoff proves the convergence of Trotter's product formula by approximating the product using the Chernoff $\\sqrt{n}$-Lemma.\n\t\tFor bounded generators, Chernoff's proof gives a convergence rate of order $n^{-\\frac{1}{2}}$. Following his proof and using our modified Chernoff Lemma, we achieve the well-known optimal convergence rate of order $n^{-1}$ \\cites[Thm.~2.11]{Hall.2015}[p.~1-2]{Neidhardt.2018}:\t\t\n\t\t\\begin{prop}[\\texorpdfstring{\\cite[Thm.~1]{Chernoff.1968}}{[6, Thm.~1]}]\\label{prop:trotter-kato-product-formula}\n\t\t\tLet $F:\\R_{\\geq0}\\rightarrow\\cB(\\cX)$ be a continuously differentiable function (in the uniform topology) satisfying $\\sup_{t\\in\\R_{\\geq0}}\\|F(t)\\|_\\infty\\leq1$. Assume that $F(0)=\\1$ and denote the derivative at $t=0$ by $\\cL\\in\\cB(\\cX)$. Then, for all $t\\geq0$\n\t\t\t\\begin{equation*}\n\t\t\t\t\\norm{F\\left(\\tfrac{t}{n}\\right)^n-e^{t\\cL}}_\\infty\\leq \\norm{n\\left(F\\left(\\tfrac{t}{n}\\right)-\\1\\right)-t\\cL}_\\infty + \\frac{n}{2}\\norm{(F\\left(\\tfrac{t}{n}\\right)-\\1)}_\\infty^2.\n\t\t\t\\end{equation*}\n\t\t\\end{prop}\n\t\t\\begin{proof}\n\t\t\tThe case $t=0$ is clear. For $t>0$, applying \\Cref{lem:improved-chernoff}, we get\n\t\t\t\\begin{align*}\n\t\t\t\t\\norm{F\\left(\\tfrac{t}{n}\\right)^n-e^{t\\cL}}_\\infty&\\leq\\norm{F\\left(\\tfrac{t}{n}\\right)^n-e^{n\\left(F\\left(\\text{\\tiny{$\\tfrac{t}{n}$}}\\right)-\\1\\right)}}_\\infty+\\norm{e^{n\\left(F\\left(\\text{\\tiny{$\\tfrac{t}{n}$}}\\right)-\\1\\right)}-e^{t\\cL}}_\\infty\\\\\n\t\t\t\t&\\leq \\frac{n}{2}\\norm{F\\left(\\tfrac{t}{n}\\right)-\\1}_\\infty^2+\\norm{e^{n\\left(F\\left(\\text{\\tiny{$\\tfrac{t}{n}$}}\\right)-\\1\\right)}-e^{t\\cL}}_\\infty.\n\t\t\t\\end{align*}\n\t\t\tFor the second term above, we apply \\Cref{lem:integral-equation-semigroups}:\n\t\t\t\\begin{align*}\n\t\t\t\t\\norm{e^{n\\left(F\\left(\\text{\\tiny{$\\tfrac{t}{n}$}}\\right)-\\1\\right)}-e^{t\\cL}}_\\infty&=\\int_{0}^{1}\\norm{e^{sn\\left(F\\left(\\text{\\tiny{$\\tfrac{t}{n}$}}\\right)-\\1\\right)}\\left(n\\left(F\\left(\\tfrac{t}{n}\\right)-\\1\\right)-t\\cL\\right)e^{(1-s)t\\cL}}_\\infty ds\\\\\n\t\t\t\t&\\leq\\norm{n\\left(F\\left(\\tfrac{t}{n}\\right)-\\1\\right)-t\\cL }_\\infty,\n\t\t\t\\end{align*}\n\t\t\twhere we use $\\|F(\\tfrac{t}{n})\\|_\\infty\\leq1$ and that $e^{sn(F(\\text{\\tiny{$\\tfrac{t}{n}$}})-\\1)}$ is a contraction semigroup.\n\t\t\\end{proof}\n\t\tApplying the proposition to the case of Trotter's product formula, we achieve the well-known optimal convergence rate for bounded generators on Banach spaces \\cites[Thm.~2.11]{Hall.2015}[p.~1-2]{Neidhardt.2018}:\n\t\t\\begin{cor}[\\texorpdfstring{\\cites[Thm.~2.11]{Hall.2015}}{[20, Thm.~2.11]}]\\label{cor:trotter-kate-product-formula-convergence}\n\t\t\tLet $\\cL_1$ and $\\cL_2$ be bounded generators of two uniformly continuous contraction semigroups. Then, for $n\\rightarrow\\infty$\n\t\t\t\\begin{equation*}\n\t\t\t\t\\norm{\\left(e^{\\frac{1}{n}\\cL_1}e^{\\frac{1}{n}\\cL_2}\\right)^n-e^{\\cL_1+\\cL_2}}_\\infty=\\cO\\left(\\frac{1}{n}\\right).\n\t\t\t\\end{equation*}\n\t\t\\end{cor}\n\t\t\\begin{proof}\n\t\t\tWe define $F(\\tfrac{1}{n})\\coloneqq e^{\\frac{1}{n}\\cL_1}e^{\\frac{1}{n}\\cL_2}$ for which \n\t\t\t\\begin{align*}\n\t\t\t\tF(\\tfrac{1}{n})-\\1&=e^{\\frac{1}{n}\\cL_1}\\left(e^{\\frac{1}{n}\\cL_2}-\\1\\right)+e^{\\frac{1}{n}\\cL_1}-\\1\\\\\n\t\t\t\t&=\\left(e^{\\frac{1}{n}\\cL_1}-\\1\\right)\\left(e^{\\frac{1}{n}\\cL_2}-\\1\\right)+e^{\\frac{1}{n}\\cL_2}-\\1+e^{\\frac{1}{n}\\cL_1}-\\1\n\t\t\t\\end{align*}\n\t\t\tholds. Moreover,\n\t\t\t\\begin{equation*}\n\t\t\t\t\\norm{e^{\\frac{1}{n}\\cL_1}-\\1}_\\infty=\\frac{1}{n}\\norm{\\int_{0}^{1}e^{\\frac{\\tau_1}{n}\\cL_1}\\cL_1d\\tau_1}_\\infty\\leq\\frac{1}{n}\\norm{\\cL_1}_\\infty\n\t\t\t\\end{equation*}\n\t\t\tand\n\t\t\t\\begin{equation*}\n\t\t\t\t\\norm{n\\left(e^{\\frac{1}{n}\\cL_1}-\\1\\right)-\\cL_1}_\\infty=\\frac{1}{n}\\norm{\\int_{0}^{1}\\int_{0}^{1}\\tau_1e^{\\frac{\\tau_1\\tau_2}{n}\\cL_1}\\cL_1^2d\\tau_2d\\tau_1}_\\infty\\leq\\frac{1}{2n}\\norm{\\cL_1}^2_\\infty.\n\t\t\t\\end{equation*}\n\t\t\t\\begin{align*}\n\t\t\t\t\\norm{n\\left(F(\\tfrac{1}{n})-\\1\\right)-\\cL_1-\\cL_2}_\\infty\\leq\\frac{1}{n}\\left(\\norm{\\cL_1}_\\infty\\norm{\\cL_2}_\\infty+2\\norm{\\cL_1}^2_\\infty+2\\norm{\\cL_2}^2_\\infty\\right)\n\t\t\t\\end{align*}\n\t\t\tand the statement follows from \\Cref{prop:trotter-kato-product-formula}.\n\t\t\\end{proof}\n\t\n\t\n\t\\section{Strongly Continuous Zeno Dynamics}\\label{sec:unbounded-generator-zeno-subspace}\n\t\tWe proceed with the statement and proof of our first main result, namely \\Cref{thm:spectral-gap}, which we restate here for sake of clarity of conciseness:\n\t\t\n\t\t\\begin{thm}\\label{thm:spectral-gap}\n\t\t Let $(\\cL,\\cD(\\cL))$ be the generator of a $C_0$-contraction semigroup on $\\cX$, $M\\in\\cB(\\cX)$ a contraction, and $P$ a projection satisfying \n\t\t\t\\begin{equation}\\label{unifpower1}\n\t\t\t\t\\norm{M^n-P}_\\infty\\leq\\delta^n\n\t\t\t\\end{equation}\n\t for $\\delta\\in(0,1)$ and all $n\\in\\N$. Moreover, assume there is $b\\geq0$ so that for all $t\\ge 0$\n\t\t\t\\begin{equation}\\label{eq:thm1-asympzeno}\n\t\t\t\t\\norm{Pe^{t\\cL}P^\\perp}_\\infty\\leq tb\\quad\\text{and}\\quad\\norm{P^\\perp e^{t\\cL}P}_\\infty\\leq tb.\n\t\t\t\\end{equation}\n\t\t\tIf $(P\\cL P,\\cD(\\cL P))$ is the generator of a $C_0$-semigroup, then for any $t\\ge 0$ and all $x\\in\\cD((\\cL P)^2)$ \n\t\t\t\\begin{equation*}\n\t\t\t\t\\norm{\\left(Me^{\\frac{t}{n}\\cL}\\right)^nx-e^{tP\\cL P}Px}\\leq\\frac{c(t,b)}{n}\\left(\\norm{x}+\\norm{\\cL P x}+\\norm{(\\cL P)^2x}\\right)+\\delta^n\\norm{x}\n\t\t\t\\end{equation*}\n\t\tfor a constant $c(t,b)>0$ depending on $t$ and $b$, but independent of $n$ .\n\t\t\\end{thm}\n\t\t\\subsection{Proof of \\texorpdfstring{\\Cref{thm:spectral-gap} }{Theorem 3.1}}\\label{subsec:proofthm1}~\\\\\n\t\t\tWe assume for sake of simplicity that $t=1$, and split our proof in three parts:\n\t\t\t\\begin{align}\n\t\t\t\t\\norm{\\left(Me^{\\frac{1}{n}\\cL}\\right)^nx-e^{P\\cL P}Px}\\leq\\;&\\norm{\\left(Me^{\\frac{1}{n}\\cL}\\right)^nx-\\left(Pe^{\\frac{1}{n}\\cL}P\\right)^nx}\\label{eq:proofthm1-term1}\\\\\t+\\;&\\norm{\\left(Pe^{\\frac{1}{n}\\cL}P\\right)^nx-e^{nP(e^{\\frac{1}{n}\\cL}-\\1)P}Px}\\label{eq:proofthm1-term2}\\\\\n\t\t\t\t+\\;&\\norm{e^{nP(e^{\\frac{1}{n}\\cL}-\\1)P}Px-e^{P\\cL P}Px}\\label{eq:proofthm1-term3}\n\t\t\t\\end{align}\n\t\t\tfor all $x\\in\\cD((\\cL P)^2)$.\n\t\t\t\n\t\t\t\\subsubsection{Upper bound on \\Cref{eq:proofthm1-term1}:}\n\t\t\t\tThe following \\namecref{lem:proofthm1-term1} uses similar proof strategies as Lemma 3 in \\cites{Burgarth.2020} and extends the result to infinite dimensions in the strong topology.\n\t\t\t\t\\begin{lem}\\label{lem:proofthm1-term1} \n\t\t\t\t\tLet $(\\cL,\\cD(\\cL))$ be the generator of a $C_0$-contraction semigroup on $\\cX$ and $M\\in\\cB(\\cX)$ a contraction satisfying the assumptions in \\Cref{thm:spectral-gap}. Then, for all $x\\in\\cX$\n\t\t\t\t\t\\begin{equation*}\n\t\t\t\t\t\t\\norm{\\left(Me^{\\frac{1}{n}\\cL}\\right)^nx-\\left(Pe^{\\frac{1}{n}\\cL}P\\right)^nx}\\leq\\left(\\delta^n+\\frac{b}{n}+\\frac{1}{n}\\frac{b(2+b)(\\delta-\\delta^{n})}{1-\\delta}e^{2b}\\right)\\norm{x}.\n\t\t\t\t\t\\end{equation*}\n\t\t\t\t\\end{lem}\n\t\t\t\tThe proof of the above \\namecref{lem:proofthm1-term1} relies on a counting method: more precisely, we need to count the number of transitions in a binary sequence. This is related to the \\textit{urn problem}, where $k$ indistinguishable balls are placed in $l$ distinguishable urns \\cite[Chap.~1.9]{Stanley.1986}. Then, there are \n\t\t\t\t\\begin{equation}\\label{eq:urn-problem}\n\t\t\t\t\t\\binom{k-1}{l-1}\n\t\t\t\t\\end{equation}\n\t\t\t\tpossibilities to distribute the balls so that each urn contains at least one ball. \n\t\t\t\t\\begin{defi}\\label{defi:counting-patterns}\n\t\t\t\t\tLet $S=\\{A,B\\}$, $j,n,k\\in\\N$, and $n\\geq1$. We define\n\t\t\t\t\t\\begin{align*}\n\t\t\t\t\t\tS_{n,k}&\\coloneqq\\{s\\in S^n\\;|\\;\\text{$A$ appears $k$ times in $s$}\\}\\\\\n\t\t\t\t\t\tN(j,n,k)&\\coloneqq\\#\\{s\\in S_{n,k}\\;|\\;\\text{$s$ includes $j$ transitions $AB$ or $BA$}\\}.\n\t\t\t\t\t\\end{align*}\n\t\t\t\t\\end{defi}\n\t\t\t\tIn words, $N(j,n,k)$ counts the number of sequences consisting of $k$ $A$'s and $n-k$ $B$'s with the restriction that $A$ changes to $B$ or vice versa $j$ times.\n\t\t\t\t\\begin{ex}\n\t\t\t\t\tLet $S=\\{A,B\\}$, $n=4$, and $k=2$. Then,\n\t\t\t\t\t\\begin{align*}\n\t\t\t\t\t\tN(0,n,k)&=\\#\\emptyset & &=0\\\\\n\t\t\t\t\t\tN(1,n,k)&=\\#\\{AABB,BBAA\\} & &=2\\\\\n\t\t\t\t\t\tN(2,n,k)&=\\#\\{ABBA,BAAB\\} & &=2\\\\\n\t\t\t\t\t\tN(3,n,k)&=\\#\\{ABAB,BABA\\} & &=2.\n\t\t\t\t\t\\end{align*}\n\t\t\t\t\\end{ex}\n\t\t\t\t\\begin{lem}\\label{lem:counting-patterns-binary-sequences}\n\t\t\t\t\tLet $S=\\{A,B\\}$ and $n,k,j\\in\\N$ with $k\\leq n$. Then,\n\t\t\t\t\t\\begin{equation*}\n\t\t\t\t\t\tN(j,n,k)=\\begin{cases}\n\t\t\t\t\t\t\t2\\binom{n-k-1}{l-1}\\binom{k-1}{l-1}& \\text{if }j=2l-1\\\\\n\t\t\t\t\t\t\t\\frac{n-2l}{l}\\binom{n-k-1}{l-1}\\binom{k-1}{l-1}& \\text{if }j=2l\\\\\n\t\t\t\t\t\t\t1_{k\\in\\{0,n\\}}& \\text{if }j=0\n\t\t\t\t\t\t\\end{cases}\n\t\t\t\t\t\\end{equation*}\n\t\t\t\t\tfor $j\\in\\{0,...,2\\min\\{k,n-k\\}-1_{2k=n}\\}$. Otherwise $N(j,n,k)=0$.\n\t\t\t\t\\end{lem}\n\t\t\t\t\\begin{proof}\n\t\t\t\t\tIf $j\\geq2\\min\\{n-k,k\\}-1_{2k=n}$, then $N(j,n,k)=0$ by \\Cref{defi:counting-patterns}. Next we assume that $j=0$, the only possible sequences are $A^n$ ($k=n$) and $B^n$ ($k=0$) so that $N(0,n,k)=1_{k\\in\\{0,n\\}}$. In the following, we assume that $1\\leq j\\leq2\\min\\{n-k,k\\}-1_{2k=n}$, then there is a $s\\in S_{n,k}$ so that $s$ includes exactly $j$ transitions $AB$ or $BA$ so that $N(j,n,k)>0$. In the odd case $j=2l-1$ for $l\\in\\{1,...,\\min\\{k,n-k\\}\\}$, the element $s$ is constructed by $l$ blocks of $A$'s and $l$ blocks of $B$'s:\n\t\t\t\t\t\\begin{align*}\n\t\t\t\t\t\ts&=\\underbrace{A...A}_{1}\\overbrace{B...B}^{1}\\underbrace{A...A}_{2}B\\quad.\\quad.\\quad.\\quad A\\overbrace{B...B}^{l-1}\\underbrace{A...A}_{l}\\overbrace{B...B}^{l},\\\\\n\t\t\t\t\t\ts&=\\underbrace{B...B}_{1}\\overbrace{A...A}^{1}\\underbrace{B...B}_{2}A\\quad.\\quad.\\quad.\\quad B\\overbrace{A...A}^{l-1}\\underbrace{B...B}_{l}\\overbrace{A...A}^{l}.\n\t\t\t\t\t\\end{align*}\n\t\t\t\t\tIdentifying these blocks with distinguishable urns and the elements $A$ and $B$ with indistinguishable balls (q.v.~\\Caref{eq:urn-problem}), the task is to count the possibilities of placing $k$ $A$'s in $l$ urns and vice versa $n-k$ $B$'s in $l$ urns with the additional assumption that each urn must contain at least one $A$ or one $B$. By changing the roles of $A$ and $B$, we get twice the number of possible combinations. Therefore, one of the \\textit{Twelvefold Ways} \\cite[Chap.~1.9]{Stanley.1986} shows\n\t\t\t\t\t\\begin{equation*}\n\t\t\t\t\t\tN(j,n,k)=2\\binom{n-k-1}{l-1}\\binom{k-1}{l-1}.\n\t\t\t\t\t\\end{equation*}\n\t\t\t\t\tIn the even case $j=2l$ for $l\\in\\{1,...,\\min\\{k,n-k\\}-1_{2k=n}\\}$, we argue similarly to the odd case. The only difference is that s is constructed by $l+1$ blocks of $A$'s and $l$ blocks of $B$'s or vice versa:\n\t\t\t\t\t\\begin{align*}\n\t\t\t\t\t\ts&=\\underbrace{A...A}_{1}\\overbrace{B...B}^{1}\\underbrace{A...A}_{2}B\\quad.\\quad.\\quad.\\quad A\\overbrace{B...B}^{l-1}\\underbrace{A...A}_{l}\\overbrace{B...B}^{l}\\underbrace{A...A}_{l+1},\\\\\n\t\t\t\t\t\ts&=\\underbrace{B...B}_{1}\\overbrace{A...A}^{1}\\underbrace{B...B}_{2}B\\quad.\\quad.\\quad.\\quad A\\overbrace{A...A}^{l-1}\\underbrace{B...B}_{l}\\overbrace{A...A}^{l}\\underbrace{B...B}_{l+1}.\n\t\t\t\t\t\\end{align*}\n\t\t\t\t\tThen the \\textit{Twelvefold Ways} \\cite[Chap.~1.9]{Stanley.1986} proves the statement by\\\\\n\t\t\t\t\t\\begin{minipage}[b]{0.9\\textwidth}\n\t\t\t\t\t\t\\begin{align*}\n\t\t\t\t\t\t\tN(j,n,k)&=\\binom{n-k-1}{l-1}\\binom{k-1}{l}+\\binom{n-k-1}{l}\\binom{k-1}{l-1}\\\\\n\t\t\t\t\t\t\t&=\\frac{n-2l}{l}\\binom{n-k-1}{l-1}\\binom{k-1}{l-1}.\n\t\t\t\t\t\t\\end{align*}\n\t\t\t\t\t\\end{minipage}\n\t\t\t\t\\end{proof}\n\t\t\t\tWith the help of this counting method, we are ready to prove \\Cref{lem:proofthm1-term1}. In what follows, we identify the couple $(A,B)$ with the product $AB$ by slight abuse of notations.\n\t\t\t\t\\begin{proof}[Proof of \\Cref{lem:proofthm1-term1}]\n\t\t\t\t\tAssume w.l.o.g.~$P\\neq 0$, then $MP=PM=P$ because for all $n\\in\\N$\n\t\t\t\t\t\\begin{equation}\\label{eq:eigenprojection}\n\t\t\t\t\t\t\\|P-PM\\|_\\infty\\leq\\|(M^{n}-P)M\\|_\\infty+\\|P-M^{n+1}\\|_\\infty\\leq\\delta^n+\\delta^{n+1}\n\t\t\t\t\t\\end{equation}\n\t\t\t\t\tand $\\norm{P}_\\infty\\leq1$ holds by a similar argument because for all $n\\in\\N$\n\t\t\t\t\t\\begin{equation}\\label{eq:projection-contraction}\n\t\t\t\t\t\t\\|P\\|_\\infty\\leq\\|M^n\\|_\\infty+\\|P-M^n\\|_\\infty\\leq 1+\\delta^n.\n\t\t\t\t\t\\end{equation}\n\t\t\t\t\tThe main idea is to split $M=P+M^\\perp$ with $M^\\perp\\coloneqq P^\\perp M$ and order the terms after expanding the following polynomial appropriately. Let $A\\coloneqq M^\\perp e^{\\frac{1}{n}\\cL}$ and $B\\coloneqq Pe^{\\frac{1}{n}\\cL}$ so that\n\t\t\t\t\t\\begin{equation}\\label{eq:proofthm1-polynomial-expansion}\n\t\t\t\t\t\t\\left((P+M^\\perp) e^{\\frac{1}{n}\\cL}\\right)^n= B^n+\\sum_{k=1}^{n-1}\\sum_{s\\in S_{n,k}}s+A^n\n\t\t\t\t\t\\end{equation}\n\t\t\t\t\twhere elements in $S_{n,k}$ are identified with sequences of concatenated operators and denoted by $s$. Then, we partition summands by the number of transitions from $A$ to $B$ or vice versa and use\n\t\t\t\t\t\\begin{equation*}\n\t\t\t\t\t\t\\norm{A}_\\infty\\le \\delta\\qquad \\text{ and }\\qquad \t\\norm{AB}_\\infty=\\norm{M^\\perp P^\\perp e^{\\frac{1}{n}\\cL}PM e^{\\frac{1}{n}\\cL}}_\\infty\\leq\\delta\\frac{b}{n}.\n\t\t\t\t\t\\end{equation*}\n\t\t\t\t\tThe number of summands with $j$ transitions is equal to $N(j,n,k)$ given by \\Cref{lem:counting-patterns-binary-sequences} for $j\\in\\{1,...,m\\}$ and $m\\coloneqq2\\min\\{k,n-k\\}-1_{2k=n}$. Then, the inequality above shows \n\t\t\t\t\t\\begin{equation*}\n\t\t\t\t\t\t\\begin{aligned}\n\t\t\t\t\t\t\t\\norm{\\left(Me^{\\frac{1}{n}\\cL}\\right)^n-\\left(Pe^{\\frac{1}{n}\\cL}\\right)^n}_\\infty&\\leq\\delta^n+\\sum_{k=1}^{n-1}\\sum_{j=1}^{m}\\delta^{k}N(j,n,k)\\left(\\frac{b}{n}\\right)^j\\\\\n\t\t\t\t\t\t\t&=\\delta^n\\begin{aligned}[t]\n\t\t\t\t\t\t\t\t&+\\sum_{k=1}^{n-1}\\sum_{l=1}^{\\ceil*{\\frac{m}{2}}}\\delta^{k}2\\binom{n-k-1}{l-1}\\binom{k-1}{l-1}\\left(\\frac{b}{n}\\right)^{2l-1}\\\\\n\t\t\t\t\t\t\t\t&+\\sum_{k=1}^{n-1}\\sum_{l=1}^{\\floor*{\\frac{m}{2}}}\\delta^{k}\\frac{b(n-2l)}{nl}\\binom{n-k-1}{l-1}\\binom{k-1}{l-1}\\left(\\frac{b}{n}\\right)^{2l-1}\n\t\t\t\t\t\t\t\\end{aligned}\\\\\n\t\t\t\t\t\t\t&\\overset{(1)}{\\leq}\\delta^n+\\frac{b(2+b)}{n}\\sum_{l=1}^{\\floor*{\\frac{n}{2}}}\\sum_{k=1}^{n-1}\\delta^{k}\\frac{n^{2l-2}}{(l-1)!^2}\\left(\\frac{b}{n}\\right)^{2l-2}\\\\\n\t\t\t\t\t\t\t&=\\delta^n+\\frac{b(2+b)}{n}\\frac{\\delta-\\delta^{n}}{1-\\delta}\\sum_{l=0}^{\\floor*{\\frac{n}{2}}-1}\\frac{b^{2l}}{l!^2}\\\\\n\t\t\t\t\t\t\t&{\\leq}\\delta^n+\\frac{1}{n}\\frac{b(2+b)(\\delta-\\delta^{n})}{1-\\delta}e^{2b}.\n\t\t\t\t\t\t\\end{aligned}\n\t\t\t\t\t\\end{equation*}\n\t\t\t\t\tIn (1) above, we used the upper bound $\\binom{n}{k}\\leq\\frac{n^k}{k!}$ to show\n\t\t\t\t\t\\begin{equation*}\n\t\t\t\t\t\t\\binom{n-k-1}{l-1}\\binom{k-1}{l-1}\\leq\\frac{n^{2l-2}}{(l-1)!^2}.\n\t\t\t\t\t\\end{equation*}\n\t\t\t\t\tAdditionally, we increase the upper index to $\\floor*{\\frac{n}{2}}$, and upper bound\n\t\t\t\t\t\\begin{equation*}\n\t\t\t\t\t\t2+\\frac{b(n-2l)}{nl}\\leq2+b.\n\t\t\t\t\t\\end{equation*}\n\t\t\t\t\tApplying the assumptions again to\n\t\t\t\t\t\\begin{equation*}\n\t\t\t\t\t\t\\left(Pe^{\\frac{1}{n}\\cL}\\right)^nx-\\left(Pe^{\\frac{1}{n}\\cL}P\\right)^nx=\\left(Pe^{\\frac{1}{n}\\cL}P\\right)^{n-1}Pe^{\\frac{1}{n}\\cL}P^\\perp x\n\t\t\t\t\t\\end{equation*}\n\t\t\t\t\tfinishes the \\namecref{lem:proofthm1-term1}:\\\\\n\t\t\t\t\t\\begin{minipage}[b]{0.9\\textwidth}\n\t\t\t\t\t\t\\begin{equation*}\n\t\t\t\t\t\t\t\\norm{\\left(Me^{\\frac{1}{n}\\cL}\\right)^nx-\\left(Pe^{\\frac{1}{n}\\cL}P\\right)^nx}\\leq\\left(\\delta^n+\\frac{b}{n}+\\frac{1}{n}\\frac{b(2+b)(\\delta-\\delta^{n})}{1-\\delta}e^{2b}\\right)\\norm{x}.\n\t\t\t\t\t\t\\end{equation*}\n\t\t\t\t\t\\end{minipage}\n\t\t\t\t\\end{proof}\n\t\t\t\t\\begin{rmk*}\n\t\t\t\t\tBy the counting method introduced above, we can approximate $(Me^{\\frac{1}{n}\\cL})^n$ by $(Pe^{\\frac{1}{n}\\cL}P)^n$, which is independent of $M^\\perp$. In previous works \\cites{Mobus.2019}{Becker.2021}, the operators considered in similar proof steps as \\Cref{eq:proofthm1-term2} and (\\ref{eq:proofthm1-term3}) depended on $M^\\perp$.\n\t\t\t\t\\end{rmk*}\n\t\t\t\n\t\t\t\\subsubsection*{Upper bound on \\Cref{eq:proofthm1-term2}:}\n\t\t\t\tIn the next step, we apply our modified Chernoff \\Cref{lem:improved-chernoff}:\n\t\t\t\t\\begin{lem}\\label{lem:proofthm1-term2}\n\t\t\t\t\tLet $(\\cL,\\cD(\\cL))$ be the generator of a $C_0$-contraction semigroup on $\\cX$ and $P\\in\\cB(\\cX)$ be a projection. Assume that both operators satisfy the same assumption as in \\Cref{thm:spectral-gap}. Then, for all $x\\in\\cD((\\cL P)^2)$\n\t\t\t\t\t\\begin{equation*}\n\t\t\t\t\t\t\\norm{\\left(Pe^{\\frac{1}{n}\\cL}P\\right)^nx-e^{nP(e^{\\frac{1}{n}\\cL}-\\1)P}Px}\\leq\\frac{1}{2n}\\left(b^2\\norm{x}+b\\norm{\\cL Px}+\\norm{(\\cL P)^2x}\\right).\n\t\t\t\t\t\\end{equation*}\n\t\t\t\t\\end{lem}\n\t\t\t\t\\begin{proof}\n\t\t\t\t\tThe proof relies on the modified Chernoff Lemma (q.v.~\\Caref{lem:improved-chernoff}) applied to the contraction $C(\\tfrac{1}{n})=Pe^{\\frac{1}{n}\\cL}P$ on $P\\cX$. Then, for all $x\\in\\cX$\n\t\t\t\t\t\\begin{equation*}\n\t\t\t\t\t\t\\norm{\\left(Pe^{\\frac{1}{n}\\cL}P\\right)^nx-e^{nP(e^{\\frac{1}{n}\\cL}-\\1)P}Px}\\leq \\frac{n}{2}\\norm{\\left(P\\left(e^{\\frac{1}{n}\\cL}-\\1\\right)P\\right)^2x}.\n\t\t\t\t\t\\end{equation*}\n\t\t\t\t\tMoreover, the asymptotic Zeno condition (\\ref{eq:thm1-asympzeno}) and the continuity of the norm imply\n\t\t\t\t\t\\begin{equation*}\n\t\t\t\t\t\t\\norm{P^\\perp\\cL Px}=\\lim\\limits_{h\\rightarrow 0}\\frac{1}{h}\\norm{P^\\perp e^{h\\cL}Px}\\leq b\\norm{x}\n\t\t\t\t\t\\end{equation*}\n\t\t\t\t\tfor all $x\\in\\cD(\\cL P)$. Hence $P^\\perp\\mathcal{L}P$ is a bounded operator with $\\|P^\\perp \\cL P\\|_\\infty\\leq b$. Next, given $x\\in\\cD(\\cL P)$, the $C_0$-semigroup properties (q.v.~\\Caref{lem:properties-semigroups}) imply\n\t\t\t\t\t\\begin{align*}\n\t\t\t\t\t\tn\\norm{\\left(P\\left(e^{\\frac{1}{n}\\cL}-\\1\\right)P\\right)^2x}&=\\norm{P\\left(e^{\\frac{1}{n}\\cL}-\\1\\right)P\\int_{0}^{1}e^{\\frac{\\tau_1}{n}\\cL}(\\1-P+P)\\cL Pxd\\tau_1}\\\\\n\t\t\t\t\t\t&\\leq\\norm{P\\left(e^{\\frac{1}{n}\\cL}-\\1\\right)P\\int_{0}^{1}e^{\\frac{\\tau_1}{n}\\cL}P\\cL Pxd\\tau_1}\\\\\n\t\t\t\t\t\t&\\quad+2\\int_{0}^{1}\\norm{Pe^{\\frac{\\tau_1}{n}\\cL}P^\\perp}_\\infty d\\tau_1\\norm{P^\\perp\\cL P}_\\infty\\norm{x}.\n\t\t\t\t\t\\end{align*}\n\t\t\t\t\tNote that $\\int_{0}^{1}e^{\\frac{\\tau_1}{n}\\cL}P\\cL Px$ belongs to $\\cD(\\cL)$ by \\Cref{lem:properties-semigroups}, but not necessarily to $\\cD(\\cL P)$. However, for all $x\\in\\cD((\\cL P)^2)$\n\t\t\t\t\t\\begin{equation*}\n\t\t\t\t\t\t\\begin{aligned}\n\t\t\t\t\t\t\tn\\norm{\\left(P\\left(e^{\\frac{1}{n}\\cL}-\\1\\right)P\\right)^2x}&\\leq\\norm{P\\left(e^{\\frac{1}{n}\\cL}-\\1\\right)P\\int_{0}^{1}e^{\\frac{\\tau_1}{n}\\cL}P\\cL Pxd\\tau_1}+\\frac{b^2}{n}\\norm{x}\\\\\n\t\t\t\t\t\t\t&\\leq\\norm{P\\left(e^{\\frac{1}{n}\\cL}-\\1\\right)\\int_{0}^{1}e^{\\frac{\\tau_1}{n}\\cL}P\\cL Pxd\\tau_1}\\\\\n\t\t\t\t\t\t\t&\\quad+\\frac{b^2}{n}\\norm{x}+\\norm{Pe^{\\frac{1}{n}\\cL}P^\\perp}_\\infty\\norm{\\cL Px}\\\\\n\t\t\t\t\t\t\t&\\leq\\frac{1}{n}\\norm{P\\int_{0}^{1}e^{\\frac{\\tau_2}{n}\\cL}\\cL\\int_{0}^{1}e^{\\frac{\\tau_1}{n}\\cL}P\\cL Pxd\\tau_1d\\tau_2}\\\\\n\t\t\t\t\t\t\t&\\quad+\\frac{b^2}{n}\\norm{x}+\\frac{b}{n}\\norm{\\cL Px}\\\\\n\t\t\t\t\t\t\t&\\leq\\frac{1}{n}\\left(b^2\\norm{x}+b\\norm{\\cL Px}+\\norm{(\\cL P)^2x}\\right),\n\t\t\t\t\t\t\\end{aligned}\n\t\t\t\t\t\\end{equation*}\n\t\t\t\t\twhich proves \\Cref{lem:proofthm1-term2}.\n\t\t\t\t\\end{proof}\n\t\t\t\t\\begin{rmk*}\n\t\t\t\t\tAs regards to the convergence rate of the quantum Zeno effect, \\Cref{lem:proofthm1-term2} constitutes our main improvement compared to the work \\cites{Becker.2021}. The modified Chernoff lemma allows to improve the convergence rate to $n^{-1}$.\n\t\t\t\t\\end{rmk*}\n\t\t\t\n\t\t\t\\subsubsection*{Upper bound on \\Cref{eq:proofthm1-term3}:}\n\t\t\t\tFinally, we prove an upper bound on \\Cref{eq:proofthm1-term3}, which can be interpreted as a modified \\textit{Dunford-Segal approximation} \\cite{Gomilko.2014}.\n\t\t\t\t\\begin{lem}\\label{lem:proofthm1-term3}\n\t\t\t\t\tLet $(\\cL,\\cD(\\cL))$ be the generator of a $C_0$-contraction semigroup on $\\cX$ and $P\\in\\cB(\\cX)$ be a projection. Assume that both operators satisfy the assumptions of \\Cref{thm:spectral-gap}. Then, for all $x\\in\\cD((\\cL P)^2)$\n\t\t\t\t\t\\begin{equation*}\n\t\t\t\t\t\t\\norm{e^{nP(e^{\\frac{1}{n}\\cL}-\\1)P}Px-e^{P\\cL P}Px}\\leq\\frac{e^{\\tilde{b}}}{2n}\\left(b^2\\norm{x}+\\norm{(\\cL P)^2 x}\\right)\n\t\t\t\t\t\\end{equation*}\n\t\t\t\t\twith $e^{\\tilde{b}}\\coloneqq\\sup_{s\\in[0,1]}\\|e^{sP\\cL P}P\\|_\\infty<\\infty$.\n\t\t\t\t\\end{lem}\n\t\t\t\t\\begin{proof}\n\t\t\t\t\tThe proof relies on the integral equation for semigroups from \\Cref{lem:integral-equation-semigroups}. We start by proving the continuity of \n\t\t\t\t\t\\begin{equation}\\label{eq:proofthm1-integral-equation-continuity}\n\t\t\t\t\t\t[0,1]\\ni s\\mapsto -e^{sP\\cL P}P\\left(nP(e^{\\frac{1}{n}\\cL}-\\1)P-P\\cL P\\right)e^{(1-s)nP(e^{\\frac{1}{n}\\cL}-\\1)P}Px.\n\t\t\t\t\t\\end{equation}\n\t\t\t\t\tSince for all $s\\in[0,1]$ and $x\\in\\cD(\\cL P)$\n\t\t\t\t\t\\begin{align*}\n\t\t\t\t\t\te^{snP(e^{\\frac{1}{n}\\cL}-\\1)P}P\\cL Px&=\\lim\\limits_{h\\rightarrow0}Pe^{snP(e^{\\frac{1}{n}\\cL}-\\1)P}\\frac{P(e^{h\\cL}-\\1)P}{h}x\\\\\n\t\t\t\t\t\t&=\\lim\\limits_{h\\rightarrow0}\\frac{P(e^{h\\cL}-\\1)P}{h}e^{snP(e^{\\frac{1}{n}\\cL}-\\1)P}Px=P\\cL Pe^{snP(e^{\\frac{1}{n}\\cL}-\\1)P}Px,\n\t\t\t\t\t\\end{align*}\n\t\t\t\t\tthe vector-valued function defined in \\Cref{eq:proofthm1-integral-equation-continuity} is equal to\n\t\t\t\t\t\\begin{equation*}\n\t\t\t\t\t\t-e^{sP\\cL P}Pe^{(1-s)nP(e^{\\frac{1}{n}\\cL}-\\1)P}\\left(nP(e^{\\frac{1}{n}\\cL}-\\1)P-P\\cL P\\right)x.\n\t\t\t\t\t\\end{equation*}\n\t\t\t\t\tand, thereby, well-defined and continuous in $s$. Therefore, \\Cref{lem:integral-equation-semigroups} gives for all $x\\in\\cD(\\cL P)$\n\t\t\t\t\t\\begin{align*}\n\t\t\t\t\t\t&e^{nP(e^{\\frac{1}{n}\\cL}-\\1)P}Px-e^{P\\cL P}Px\\\\\n\t\t\t\t\t\t&\\qquad\\quad=-\\int_{0}^{1}e^{sP\\cL P}Pe^{(1-s)nP(e^{\\frac{1}{n}\\cL}-\\1)P}\\left(nP(e^{\\frac{1}{n}\\cL}-\\1)P-P\\cL P\\right)xds.\n\t\t\t\t\t\\end{align*}\n\t\t\t\t\tMoreover, for all $x\\in\\cD((\\cL P)^2)$\n\t\t\t\t\t\\begin{equation}\\label{eq:approx-generator}\n\t \t\t\t\t\\begin{aligned}\n\t \t\t\t\t nP(e^{\\frac{1}{n}\\cL}-\\1)Px-P\\cL Px&=P\\int_{0}^{1}e^{\\frac{\\tau_1}{n}\\cL}\\cL Px d\\tau_1-P\\cL Px\\\\\n\t \t\t\t\t &=\\frac{1}{n}P\\int_{0}^{1}\\int_{0}^{1}\\tau_1e^{\\frac{\\tau_1\\tau_2}{n}\\cL}(\\cL P)^2xd\\tau_2d\\tau_1+P\\int_{0}^{1}e^{\\frac{\\tau_1}{n}\\cL}P^\\perp\\cL Px d\\tau_1.\n\t\t\t\t\t \\end{aligned}\n\t\t\t\t\t\\end{equation}\n\n\t\t\t\t\tFinally, we use $\\sup_{s\\in[0,1]}\\|e^{sP\\cL P}P\\|_\\infty<\\infty$, which holds by the \\textit{principle of uniform boundedness} (q.v.~proof of \\Caref{prop:trotter-kato-product-formula}), the property that $(e^{s nP(e^{\\frac{1}{n}\\cL}-\\1)P})_{s\\geq0}$ is a contraction, and the upper bounds $\\|P^\\perp\\cL P\\|_\\infty\\leq b$ and $\\|Pe^{s\\cL}P^\\perp\\|_\\infty\\leq sb$ so that\n\t\t\t\t\t\\begin{align*}\n\t\t\t\t\t\t\\norm{e^{nP(e^{\\frac{1}{n}\\cL}-\\1)P}Px-e^{P\\cL P}Px}&\\leq e^{\\tilde{b}}\\int_{0}^{1}\\norm{\\left(nP(e^{\\frac{1}{n}\\cL}-\\1)P-P\\cL P\\right)x}ds\\\\\n\t\t\t\t\t\t&\\leq\\frac{e^{\\tilde{b}}}{2n}\\left(b^2\\|x\\|+\\|(\\cL P)^2 x\\|\\right)\n\t\t\t\t\t\\end{align*} \n\t\t\t\t\tfor all $x\\in\\cD((\\cL P)^2)$ and $e^{\\tilde{b}}=\\sup_{s\\in[0,1]}\\|e^{sP\\cL P}P\\|_\\infty$.\n\t\t\t\t\\end{proof}\n\t\t\t\tThe above approximation of $e^{P\\cL P}$ by $e^{nP(e^{\\frac{1}{n}\\cL}-\\1)P}$ is similar to the Dunford-Segal approximation, which would be given by $\\operatorname{exp}\\big({n(\\operatorname{exp}({\\frac{1}{n}P\\cL P})-\\1)}\\big)$: for the generator $(\\cK,\\cD(\\cK))$ of a bounded $C_0$-semigroup, \\citeauthor{Gomilko.2014} proved \\cite[Cor.~1.4]{Gomilko.2014}\n\t\t\t\t\\begin{equation*}\n\t\t\t\t\t\\norm{e^{nt(e^{\\frac{1}{n}\\cK}-\\1)}x-e^{t\\cK}x}\\leq8\\tilde{b}\\frac{t}{n}\\norm{\\cK^2x}\n\t\t\t\t\\end{equation*}\n\t\t\t\tfor all $x\\in\\cD(\\cK^2)$ and $\\tilde{b}\\coloneqq\\sup_{t\\geq0}\\|e^{t\\cK}\\|_\\infty$. In our case, it is not clear whether $(e^{sP\\cL P})_{s\\geq0}$ is uniformly bounded.\n\t\t\t\t\\begin{rmk*}\n\t\t\t\t\tThe specificity of the last step stems from the fact that $(\\cL,\\cD(\\cL))$ is unbounded. In the previous works \\cites{Mobus.2019}{Becker.2021} a similar step exits but in both papers $\\cL$ was assumed to be bounded. Moreover, \\Cref{eq:approx-generator} is the only step in the proof of \\Cref{thm:spectral-gap}, which deals with the operator $P\\cL P$. If $P\\cL P$ is closable, $P\\cL Px=\\overline{P\\cL P}x$ for all $x\\in\\cD(\\cL P)$ so that it is enough to ask for the closure of $P\\cL P$ to define a generator. The same reasoning works for \\Cref{prop:spectral-gap-uniform-norm-power-convergence} and \\Cref{cor:spectral-gap}.\n\t\t\t\t\\end{rmk*}\n\t\t\t\n\t\t\t\\subsubsection*{End of the proof of \\Cref{thm:spectral-gap}:}\n\t\t\t\tWe combine \\Cref{lem:proofthm1-term1}, \\ref{lem:proofthm1-term2}, and \\ref{lem:proofthm1-term3} to prove \\Cref{thm:spectral-gap}.\n\t\t\t\t\\begin{proof}[Proof of \\Cref{thm:spectral-gap}]\n\t\t\t\t\tLet $x\\in\\cD((\\cL P)^2)$. Then,\n\t\t\t\t\t\\begin{flalign*}\n\t\t\t\t\t\t&&\\norm{\\left(Me^{\\frac{1}{n}\\cL}\\right)^nx-e^{P\\cL P}Px}&\\leq\\left(\\delta^n+\\frac{b}{n}+\\frac{1}{n}\\frac{b(2+b)(\\delta-\\delta^{n})}{1-\\delta}e^{2b}\\right)\\norm{x}&&\\text{(\\Caref{lem:proofthm1-term1})}\\\\\n\t\t\t\t\t\t&& &+\\frac{1}{2n}\\left(b^2\\norm{x}+b\\norm{\\cL Px}+\\norm{(\\cL P)^2x}\\right)&&\\text{(\\Caref{lem:proofthm1-term2})}\\\\\n\t\t\t\t\t\t&& &+\\frac{e^{\\tilde{b}}}{2n}\\left(b^2\\norm{x}+\\norm{(\\cL P)^2 x}\\right)&&\\text{(\\Caref{lem:proofthm1-term3})}.\n\t\t\t\t\t\\end{flalign*}\n\t\t\t\t\tRedefining $\\cL$ by $t\\cL$ and $b$ by $tb$, we achieve\n\t\t\t\t\t\\begin{equation*}\n\t\t\t\t\t\t\\norm{\\left(Me^{\\frac{t}{n}\\cL}\\right)^nx-e^{tP\\cL P}Px}\\leq\\frac{c}{n}\\left(\\norm{x}+\\norm{\\cL Px}+\\norm{(\\cL P)^2x}\\right)+\\delta^n\\norm{x}\n\t\t\t\t\t\\end{equation*}\n\t\t\t\t\twith an appropriate constant $c>0$ and $e^{\\tilde{b}}=\\sup_{s\\in[0,t]}\\|e^{sP\\cL P}\\|_\\infty$.\n\t\t\t\t\\end{proof}\n\t\t\t\t\\begin{rmk*}\n\t\t\t\t\tThe upper bound in \\Cref{thm:spectral-gap} can be formulated for all $x\\in\\cD(\\cL P)$. For this, one must stop at an earlier stage of the proof and express the error terms by appropriate integrals. One possible bound would be the following \n\t\t\t\t\t\\begin{flalign*}\n\t\t\t\t\t\t&& \\|\\left(Me^{\\frac{1}{n}\\cL}\\right)^nx&-e^{P\\cL P}Px\\| &&\\\\\n\t\t\t\t\t\t&& &\\leq\\left(\\delta^n+\\frac{b}{n}+\\frac{1}{n}\\frac{b(2+b)(\\delta-\\delta^{n})}{1-\\delta}e^{2b}\\right)\\norm{x}&&\\text{(\\Caref{lem:proofthm1-term1})}\\\\\n\t\t\t\t\t\t&& &+\\frac{1}{2n}\\left(b^2\\norm{x}+b\\norm{\\cL Px}+\\norm{P\\int_{0}^{1}\\int_{0}^{1}\\cL e^{\\frac{\\tau_1+\\tau_2}{n}\\cL}P\\cL Pxd\\tau_1d\\tau_2}\\right)&&\\text{(\\Caref{lem:proofthm1-term2})}\\\\\n\t\t\t\t\t\t&& &+\\frac{e^{\\tilde{b}}}{2n}\\left(b^2\\norm{x}+2\\norm{P\\int_{0}^{1}\\int_{0}^{1}\\tau_1\\cL e^{\\frac{\\tau_1\\tau_2}{n}\\cL}P\\cL Pxd\\tau_2d\\tau_1}\\right)&&\\text{(\\Caref{lem:proofthm1-term3})}.\n\t\t\t\t\t\\end{flalign*}\n\t\t\t\t\\end{rmk*}\n\t\t\t\\subsection{Proof of \\texorpdfstring{\\Cref{cor:explicit-bound-thm1} and \\Cref{cor:explicit-bound-thm1-closed-sys}}{Proposition 3.1 and Corollary 3.2}}\\label{subsec:proofexplicitboundthm1}\n\t\t\t\t\\begin{proof}[Proof of \\Cref{cor:explicit-bound-thm1}]\n\t\t\t\t\tSince $\\|P\\|_\\infty\\leq1$ (\\ref{eq:projection-contraction}) and $t\\mapsto e^{t\\cL}$ is a uniformly continuous contraction semigroup, the generator is defined on $\\cX$ and bounded, i.e.~$\\|\\cL\\|_\\infty<\\infty$, so that\n\t\t\t\t\t\\begin{equation*}\n\t\t\t\t\t\t\\norm{Pe^{t\\cL}(\\1-P)}_\\infty=\\norm{P(e^{t\\cL}-\\1)(\\1-P)}_\\infty\\leq t\\norm{\\cL\\int_0^1e^{ts\\cL}(\\1-P)ds}_\\infty\\leq t\\norm{\\cL}_\\infty c_{p}.\n\t\t\t\t\t\\end{equation*}\n\t\t\t\t\twhere $c_{p}\\coloneqq\\|\\1-P\\|_\\infty\\leq2$. Then, we simplify the bounds found in \\Cref{lem:proofthm1-term1}, \\ref{lem:proofthm1-term2}, and \\ref{lem:proofthm1-term3} to\n\t\t\t\t\t\\begin{equation*}\n\t\t\t\t\t\t\\norm{\\left(Me^{\\frac{1}{n}\\cL}\\right)^n-e^{P\\cL P}P}_\\infty\\leq\\delta^n+\\frac{1}{n}\\left(b+\\frac{2\\delta}{1-\\delta}e^{3b}+\\frac{b^2}{2}+\\frac{b}{2}\\norm{\\cL}_\\infty+\\frac{1}{2}\\norm{\\cL}^2_\\infty+\\frac{e^{\\tilde{b}}b^2}{2}+\\frac{e^{\\tilde{b}}}{2}\\norm{\\cL}^2_\\infty\\right),\n\t\t\t\t\t\\end{equation*}\n\t\t\t\t\twhere $e^{\\tilde{b}}=\\sup_{s\\in[0,t]}\\|e^{sP\\cL P}\\|_\\infty$. By redefining $\\cL$ by $t\\cL$, $b$ by $tb$, and using $b\\leq\\|\\cL\\|_\\infty c_{p}$,\n\t\t\t\t\t\\begin{align*}\n\t\t\t\t\t\t\\norm{\\left(Me^{\\frac{t}{n}\\cL}\\right)^n-e^{tP\\cL P}P}_\\infty\\leq c_{p}\\frac{t\\|\\cL\\|_\\infty}{n} &+\\left(\\frac{c_{p}+(1+e^{\\tilde{b}})(1+c_{p}^2)}{2}\\right)\\frac{t^2\\|\\cL\\|_\\infty^2}{n}\\\\\n\t\t\t\t\t\t&+\\delta^n+\\frac{2\\delta}{1-\\delta}\\frac{e^{3t\\|\\cL\\|_\\infty c_{p}}}{n}\n\t\t\t\t\t\\end{align*}\n\t\t\t\t\twhich proves the statement.\n\t\t\t\t\\end{proof}\n\t\t\t\t\\begin{proof}[Proof of \\Cref{cor:explicit-bound-thm1-closed-sys}]\n\t\t\t\t\tIn closed quantum systems $\\cX=\\cH$ equipped with the operator norm induced by the scalar product, which shows $\\|U\\|_\\infty=1$ for all unitaries $U\\in\\cB(\\cH)$. Especially, $e^{\\tilde{b}}=\\sup_{s\\in[0,t]}\\|e^{sP\\cL P}\\|_\\infty=1$ because $\\|P\\|_\\infty=1$ is equivalent to $P=P^\\dagger$ \\cite[Thm.~2.1.9]{Simon.2015} so that $PHP$ is hermitian. Moreover, $P=P^\\dagger$ implies $(\\1-P)^\\dagger=(\\1-P)$ which shows $c_{p}=\\|\\1-P\\|_\\infty\\leq1$. Finally, the choice $M=P$ implies $\\delta=0$ which proves the \\nameCref{cor:explicit-bound-thm1-closed-sys} by inserting the constants into \\Cref{cor:explicit-bound-thm1}.\n\t\t\t\t\\end{proof}\n\t\t\t\\subsection{Proof of \\texorpdfstring{\\Cref{prop:spectral-gap-uniform-norm-power-convergence} }{Proposition 5.7}}\\label{subsec:thm1-prop}~\\\\\n\t\t\t\tIn this subsection, we weaken the assumptions (\\ref{unifpower1}) on the contraction $M$ at the cost of stronger assumptions on the $C_0$-semigroup. For that, we combine techniques from holomorphic functional calculus with the semicontinuity of the spectrum of $M$ perturbed by the semigroup under certain conditions. We refer to \\Cref{sec:appendix-holomorphic-fc-semicontinuity} for details on the tools needed to prove the main result of this section.\n\t\t\t\t\\begin{prop}\\label{prop:spectral-gap-uniform-norm-power-convergence}\n\t\t\t \tLet $(\\cL,\\cD(\\cL))$ be the generator of a $C_0$-contraction semigroup on $\\cX$ and $M\\in\\cB(\\cX)$ a contraction such that\n\t\t\t \t\\begin{equation}\\label{unifpowerc}\n\t\t\t \t\t\\norm{M^n-P}_\\infty\\leq\\tilde{c}\\,\\delta^n\n\t\t \t\t\\end{equation}\n\t \t\t\tfor some projection $P$, $\\delta\\in(0,1)$ and $\\tilde{c}\\ge 0$. Moreover, we assume that there is $b\\geq0$ so that \n\t \t\t\t\\begin{equation}\\label{key}\n\t \t\t\t\t\\norm{Pe^{t\\cL}P^\\perp}_\\infty\\leq tb,\\quad\\norm{M^\\perp e^{t\\cL}-M^\\perp}_\\infty\\leq tb,\\quad\\text{and}\\quad\\norm{P^\\perp e^{t\\cL}P}_\\infty\\leq tb\n \t\t\t\t\\end{equation}\n \t\t\t\twhere $M^\\perp=(\\1-P)M$. If $(P\\cL P,\\cD(\\cL P))$ generates a $C_0$-semigroup, then there is $\\epsilon>0$ such that for all $t\\geq0$, $n\\in\\N$ satisfying $t\\in[0,n\\epsilon]$, $\\tilde{\\delta}\\in(\\delta,1)$, and $x\\in\\cD((\\cL P)^2)$ \n\t\t\t\t\t\\begin{equation*}\n\t\t\t\t\t\t\\norm{\\left(Me^{\\frac{t}{n}\\cL}\\right)^nx-e^{tP\\cL P}Px}\\leq\\frac{c_1(t,b,\\tilde{\\delta}-\\delta)}{n}\\left(\\norm{x}+\\norm{\\cL P x}+\\norm{(\\cL P)^2x}\\right)+c_2(\\tilde{c},\\tilde{\\delta}-\\delta)\\tilde{\\delta}^n\\norm{x}\\,,\n\t\t\t\t\t\\end{equation*}\n\t\t\t\t\tfor some constants $c_1,c_2\\ge 0$ depending on $t$, $b$, the difference $\\tilde{\\delta}-\\delta$, and $\\tilde{c}$.\n\t\t\t\t\\end{prop}\n\t\t\t\tThe only difference to the proof of \\Cref{thm:spectral-gap} is summarized in the question: How can we upper bound $\\|(M^\\perp e^{\\frac{t}{n}\\cL})^k\\|_\\infty$ for all $k\\in\\{1,...,n\\}$ with the weaker assumption (\\ref{unifpowerc}) on $M$? For that, we replace the argument in the proof of \\Cref{lem:proofthm1-term1}, which only works for the case $\\tilde{c}=1$.\n\t\t\t\t\\begin{proof}\n\t\t\t\t\tSince the bounds found in \\Cref{lem:proofthm1-term2} and \\ref{lem:proofthm1-term3} are independent of the value of $\\tilde{c}$, it is enough to improve \\Cref{lem:proofthm1-term1}:\n\t\t\t\t\t\\begin{flalign*}\n\t\t\t\t\t\t&&\\norm{\\left(Me^{\\frac{t}{n}\\cL}\\right)^nx-e^{tP\\cL P}Px}&\\leq\\norm{\\left(Me^{\\frac{t}{n}\\cL}\\right)^nx-\\left(Pe^{\\frac{t}{n}\\cL}P\\right)^nx}&&\\\\\n\t\t\t\t\t\t&& &+\\frac{t^2}{2n}\\left(b^2\\norm{x}+b\\norm{\\cL Px}+\\norm{(\\cL P)^2x}\\right)&&\\text{(\\Caref{lem:proofthm1-term2})}\\\\\n\t\t\t\t\t\t&& &+\\frac{e^{\\tilde{b}}t^2}{2n}\\left(b^2\\norm{x}+\\norm{(\\cL P)^2 x}\\right)&&\\text{(\\Caref{lem:proofthm1-term3})}\n\t\t\t\t\t\\end{flalign*}\n\t\t\t\t\twith $e^{\\tilde{b}}=\\sup_{s\\in[0,t]}\\|e^{sP\\cL P}\\|_\\infty<\\infty$. Since the assumption (\\ref{unifpowerc}) on $M$ is a special case of the uniform power convergence (q.v.~\\Caref{unifpower}), \\Cref{prop:equivalence-spectral-gap} shows the equivalence of the uniform\\\\[1ex]\n\t\t\t\t\t\\begin{minipage}[c]{0.69\\textwidth}\n\t\t\t\t\t\tpower convergence of $M$ to the spectral gap assumption, that is\n\t\t\t\t\t\t\\begin{equation*}\n\t\t\t\t\t\t\t\\sigma(M)\\subset\\D_\\delta\\cup\\{1\\},\n\t\t\t\t\t\t\\end{equation*}\n\t\t\t\t\t\twhere the quasinilpotent operator corresponding to the eigenvalue $1$ vanishes. Therefore, the eigenprojection $P$ w.r.t.~$1$ satisfies $MP=PM=P$ and the curve \t$\\gamma:[0,2\\pi]\\rightarrow\\C,\\varphi\\mapsto\\tilde{\\delta}e^{i\\varphi}$ encloses the spectrum of $M^\\perp\\coloneqq MP^\\perp$ (q.v.~\\Caref{fig3}). Together with the second bound in \\eqref{key}, \\Cref{lem:semicontinuity-spectrum} shows that there exists $\\epsilon>0$ so that the spectrum of $M^\\perp e^{s\\cL}$ can be separated by $\\gamma$ for all $s\\in[0,\\epsilon]$.\n\t\t\t\t\t\\end{minipage}\n\t\t\t\t\t\\begin{minipage}[c]{0.3\\textwidth}\n\t\t\t\t\t\t\\begin{center}\n\t\t\t\t\t\t\t\\begin{tikzpicture}\n\t\t\t\t\t\t\t\t\\draw[dashed,tumivory] (0,0) ellipse (1.4cm and 1.4cm);\n\t\t\t\t\t\t\t\t\\draw[dashed,mygreen] (0,0) ellipse (1.15cm and 1.15cm);\n\t\t\t\t\t\t\t\t\\filldraw (1.4,0) ellipse (0.03cm and 0.03cm);\n\t\t\t\t\t\t\t\t\\filldraw (0,0) ellipse (0.03cm and 0.03cm);\n\t\t\t\t\t\t\t\t\\draw (1.7,0) node {$1$};\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\\draw[<->] (0.0,0.05)--(0,0.95);\n\t\t\t\t\t\t\t\t\\draw[<->,mygreen] (0.0433,-0.025)--(0.9526cm,-0.55cm);\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\\fill[pattern=my north east lines] (0,0) ellipse (1cm and 1cm);\n\t\t\t\t\t\t\t\t\\filldraw[white] (-0.35,0.5) ellipse (0.15cm and 0.25cm);\n\t\t\t\t\t\t\t\t\\draw (-0.35,0.5) node {$\\delta$};\n\t\t\t\t\t\t\t\t\\filldraw[white] (0.35,-0.5) ellipse (0.15cm and 0.25cm);\n\t\t\t\t\t\t\t\t\\draw[mygreen] (0.35,-0.5) node {$\\tilde{\\delta}$};\n\t\t\t\t\t\t\t\t\\filldraw[white] (-1.06,-1.06) ellipse (0.15cm and 0.3cm);\n\t\t\t\t\t\t\t\t\\draw (-1.06,-1.06) node [mygreen]{$\\gamma$};\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\\draw (1.5,1.2) node {$\\sigma(M)$};\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\\end{tikzpicture}\\\\[1ex]\n\t\t\t\t\t\t\t\\captionof{figure}{}\\label{fig3}\n\t\t\t\t\t\t\\end{center}\n\t\t\t\t\t\\end{minipage}\n\t\t\t\t\tTherefore, the holomorphic functional calculus (q.v.~\\Caref{prop:holomorphic-functional-calculus}) shows for all $t\\in[0,n\\epsilon], k\\in\\N$\n\t\t\t\t\t\\begin{equation}\\label{eq:proofthm1-upper-bound-inner-part}\n\t\t\t\t\t\t\\left(M^\\perp e^{\\frac{t}{n}\\cL}\\right)^{k}=\\frac{1}{2\\pi i}\\oint_{\\gamma}z^{k}R(z,M^\\perp e^{\\frac{t}{n}})dz.\n\t\t\t\t\t\\end{equation}\n\t\t\t\t\tLet $t\\geq0$, $n\\in\\N$ so that $t\\in[0,n\\epsilon]$. By the \\textit{principle of stability of bounded invertibility} \\cite[Thm.~IV.2.21]{Kato.1995}, $R(z,M^\\perp e^{s\\cL})$ is well-defined and bounded for all $z\\in\\gamma$ and $s\\in[0,\\epsilon]$. More explicitly, using the \\textit{second Neumann series} for the resolvent \\cite[p.~67]{Kato.1995}, we have \n\t\t\t\t\t\\begin{equation}\\label{eq:proofthm1-perturbed-resolvent-uniform-boundedness}\n\t\t\t\t\t\t\\begin{aligned}\n\t\t\t\t\t\t\t\\norm{R(z,M^\\perp e^{s\\cL})}_\\infty&=\\norm{R(z,M^{\\perp})\\sum_{p=0}^{\\infty}\\left((M^\\perp e^{s\\cL}-M^\\perp)R(z,M^\\perp)\\right)^p}_\\infty\\\\\n\t\t\t\t\t\t\t&\\leq\\norm{R(z,M^\\perp)}_\\infty\\sum_{p=0}^{\\infty}\\left(sb\\norm{R(z,M^\\perp)}_\\infty\\right)^p\\\\\n\t\t\t\t\t\t\t&\\leq\\sup_{z\\in\\gamma}\\norm{R(z,M^\\perp)}_\\infty \\frac{2+2\\tilde{\\delta}^2}{1+2\\tilde{\\delta}^2}\\eqqcolon c_2,\n\t\t\t\t\t\t\\end{aligned}\n\t\t\t\t\t\\end{equation}\n\t\t\t\t\twhere we have applied the assumption \\eqref{key} and the following upper bound on $s$ (q.v.~\\Caref{eq:semicontinuity-time-bound}):\n\t\t\t\t\t\\begin{equation*}\n\t\t\t\t\t\ts\\leq\\epsilon<\\frac{1}{2b}(1+\\tilde{\\delta}^2)^{-1}\\left(1+\\sup_{z\\in\\gamma}\\norm{R(z,M^\\perp)}_\\infty^2\\right)^{-\\frac{1}{2}}\\leq\\frac{1}{2b}(1+\\tilde{\\delta}^2)^{-1}\\left(\\sup_{z\\in\\gamma}\\norm{R(z,M^\\perp)}_\\infty\\right)^{-1},\n\t\t\t\t\t\\end{equation*}\n\t\t\t\t\tto compute the geometric series. Combining \\Cref{eq:proofthm1-upper-bound-inner-part} and (\\ref{eq:proofthm1-perturbed-resolvent-uniform-boundedness}) shows for all $k\\in\\{1,...,n\\}$\n\t\t\t\t\t\\begin{equation}\\label{eq:proofprop-perturbed-measurement}\n\t\t\t\t\t\t\\norm{\\left(M^\\perp e^{\\frac{t}{n}\\cL}\\right)^{k-1}}_\\infty\\leq\\frac{1}{2\\pi}\\oint_{\\gamma}\\abs{z}^{k-1}\\norm{R(z,M^\\perp e^{\\frac{t}{n}\\cL})}_\\infty dz\\leq c_2\\tilde{\\delta}^{k}.\n\t\t\t\t\t\\end{equation}\n\t\t\t\t\tNext, we define $A\\coloneqq M^\\perp e^{\\frac{{t}}{n}\\cL}$ and $B\\coloneqq Pe^{\\frac{{t}}{n}\\cL}$ and expand\n\t\t\t\t\t\\begin{equation*}\n\t\t\t\t\t\t\\left(Me^{\\frac{t}{n}\\cL}\\right)^n=\\left(PMe^{\\frac{t}{n}\\cL}+M^\\perp e^{\\frac{t}{n}\\cL}\\right)^n=(B+A)^n.\n\t\t\t\t\t\\end{equation*}\n\t\t\t\t\tThe above $n^{\\text{th}}$-power can be expanded in terms of sequences of the form \n\t\t\t\t\t\\begin{equation*}\n\t\t\t\t\t\tA...AB...BA...AB...\\quad\\text{or}\\quad B...BA...AB...BA...\n\t\t\t\t\t\\end{equation*}\n\t\t\t\t\tSimilarly to \\Cref{lem:proofthm1-term1}, we can upper bound every sequence w.r.t.~the number of transitions $AB$ or $BA$ using the assumptions \\eqref{key} on the $C_0$-semigroup as well as the inequality (\\ref{eq:proofprop-perturbed-measurement}). The only difference to the proof of \\Cref{lem:proofthm1-term1} is the constant $c_2$ in the inequality so that\n\t\t\t\t\t\\begin{equation*}\n\t\t\t\t\t\t\\begin{aligned}\n\t\t\t\t\t\t\t\\norm{\\left(Me^{\\frac{t}{n}\\cL}\\right)^n-\\left(Pe^{\\frac{t}{n}\\cL}\\right)^n}_\\infty&\\leq c_2\\tilde{\\delta}^n+\\sum_{k=1}^{n-1}\\sum_{j=1}^{m}\\tilde{\\delta}^{k}N(j,n,k)\\left(\\frac{tbc_2}{n}\\right)^j\\\\\n\t\t\t\t\t\t\t&\\leq c_2\\tilde{\\delta}^n+\\frac{b}{n}+\\frac{1}{n}\\frac{bc_2(2+bc_2)(\\tilde{\\delta}-\\tilde{\\delta}^{n})}{1-\\tilde{\\delta}}e^{2bc_2},\n\t\t\t\t\t\t\\end{aligned}\n\t\t\t\t\t\\end{equation*}\n\t\t\t\t\twhere $m\\coloneqq2\\min\\{k,n-k\\}-1_{2k=n}$. Then, for all $x\\in\\cD((\\cL P)^2)$ and an appropriate $c_1\\geq0$\\\\\n\t\t\t\t\t\\begin{minipage}[b]{0.9\\textwidth}\n\t\t\t\t\t\t\\begin{equation*}\n\t\t\t\t\t\t\t\\norm{\\left(Me^{\\frac{t}{n}\\cL}\\right)^nx-e^{tP\\cL P}Px}\\leq\\frac{c_1}{n}\\left(\\norm{x}+\\norm{\\cL Px}+\\norm{(\\cL P)^2x}\\right)+c_2{\\tilde{\\delta}}^n\\norm{x}.\n\t\t\t\t\t\t\\end{equation*}\n\t\t\t\t\t\\end{minipage}\n\t\t\t\t\\end{proof}\n\t\t\t\t\\vspace{-2ex}\n\t\t\t\t\\begin{proof}[Proof of \\Cref{cor:explicit-bound-prop}]\n\t\t\t\t\tSimilarly to \\Cref{cor:explicit-bound-thm1},\n\t\t\t\t\t\\begin{equation*}\n\t\t\t\t\t\t\\norm{\\left(Me^{\\frac{t}{n}\\cL}\\right)^n-e^{tP\\cL P}P}_\\infty\\leq c_2\\tilde{\\delta}^n+\\frac{tc_{p}\\|\\cL\\|_\\infty}{n} +\\frac{c_{p}+(1+e^{\\tilde{b}})(1+c_{p}^2)}{2}\\frac{t^2\\|\\cL\\|_\\infty^2}{n}+\\frac{2\\tilde{\\delta}}{1-\\tilde{\\delta}}\\frac{e^{3tc_{p}c_2\\|\\cL\\|_\\infty}}{n}\n\t\t\t\t\t\\end{equation*}\n\t\t\t\t\twhere $e^{\\tilde{b}}=\\sup_{s\\in[0,t]}\\|e^{sP\\cL P}\\|_\\infty$ and $c_2\\coloneqq\\sup_{z\\in\\gamma}\\norm{R(z,M^\\perp)}_\\infty \\frac{2+2\\tilde{\\delta}^2}{1+2\\tilde{\\delta}^2}$ (see \\Caref{eq:proofthm1-perturbed-resolvent-uniform-boundedness}).\n\t\t\t\t\tThe constant $c_2$ can be bounded with the help of the \\textit{first von Neumann series} \\cite[p.~37]{Kato.1995} and the geometric series:\n\t\t\t\t\t\\begin{equation*}\n\t\t\t\t\t\t\\frac{2+2\\tilde{\\delta}^2}{1+2\\tilde{\\delta}^2}\\sup\\limits_{z\\in\\gamma}\\norm{R(z,M^\\perp)}_\\infty=2\\sup\\limits_{z\\in\\gamma}\\|\\sum_{k=0}^{\\infty}z^{-(k+1)}(M^\\perp)^k\\|_\\infty\\leq2\\frac{\\tilde{c}}{\\tilde{\\delta}}\\sum_{k=0}^{\\infty}\\tilde{\\delta}^{-k}\\delta^k=\\frac{2\\tilde{c}}{\\tilde{\\delta}-\\delta}\n\t\t\t\t\t\\end{equation*}\n\t\t\t\t\tso that \n\t\t\t\t\t\\begin{align*}\n\t \t \\norm{\\left(Me^{\\frac{t}{n}\\cL}\\right)^n-e^{tP\\cL P}P}_\\infty\\leq\\frac{tc_{p}\\|\\cL\\|_\\infty}{n}&+\\frac{c_{p}+(1+e^{\\tilde{b}})(1+c_{p}^2)}{2}\\frac{t^2\\|\\cL\\|_\\infty^2}{n}\\\\\n\t \t &+\\frac{2\\tilde{c}}{\\tilde{\\delta}-\\delta}\\tilde{\\delta}^n+\\frac{2\\tilde{\\delta}}{1-\\tilde{\\delta}}\\frac{e^{\\frac{6tc_{p}\\tilde{c}\\|\\cL\\|_\\infty}{\\tilde{\\delta}-\\delta}}}{n}\n\t \t\\end{align*}\n \t\twhich finishes the proof of the \\namecref{cor:explicit-bound-prop}.\n\t\t\t\t\\end{proof}\n\t\t\n\t\\section{Uniform Power Convergence with Finitely Many Eigenvalues}\\label{sec:finitely-many-eigenvalues}\n\t\tIn this section, we weaken the assumption on $M$ to the uniform power convergence assumption (q.v.~\\Caref{unifpower}), that is we allow for finitely many eigenvalues $\\{\\lambda_j\\}_{j=1}^J$ and associated projections $\\{P_j\\}_{j=1}^J$ satisfying $P_jP_k=1_{j=k}P_j$. Similarly to Theorem 3 in \\cite{Becker.2021}, we strengthen the assumptions on the $C_0$-semigroup to $M\\cL$ and $\\cL P_\\Sigma$ being densely defined and bounded by $b\\geq0$, where $P_\\Sigma\\coloneqq \\sum_{j=1}^{J}P_j$. Under those assumptions, we can prove the Zeno convergence in the uniform topology:\n\n\t\t\\begin{thm}\\label{thm:spectral-gap-uniform}\n\t\t Let $(\\cL,\\cD(\\cL))$ be the generator of a $C_0$-contraction semigroup on $\\cX$ and $M\\in\\cB(\\cX)$ a contraction satisfying the following uniform power convergence: there is $\\tilde{c}>0$ so that\n\t\t\t\\begin{equation}\\label{unifpower}\n\t\t\t \\norm{M^n-\\sum_{j=1}^{J}\\lambda^n_jP_j}_\\infty\\leq\\tilde{c}\\,\\delta^n\n\t\t \\end{equation}\n\t\t for a set of projections $\\{P_j\\}_{j=1}^J$ satisfying $P_jP_k=1_{j=k}P_j$, eigenvalues $\\{\\lambda_j\\}_{j=1}^J\\subset\\partial\\D_1$, and a rate $\\delta\\in(0,1)$. For $P_\\Sigma\\coloneqq\\sum_{j=1}^{J}P_j$, we assume that $M\\cL$ and $\\cL P_\\Sigma$ are densely defined and bounded by $b\\geq0$ and $\\|P_j\\|_\\infty=1$ for all $j\\in\\{1,...,J,\\Sigma\\}$. Then, there is an $\\epsilon>0$ such that for all $n\\in\\N$, $t\\geq0$ satisfying $t\\in[0,n\\epsilon]$, and $\\tilde{\\delta}\\in(\\delta,1)$\n\t\t\t\\begin{equation*}\n\t\t\t\t\\norm{\\left(Me^{\\frac{t}{n}\\cL}\\right)^n-\\sum_{j=1}^{J}\\lambda_j^ne^{tP_j\\cL P_j}P_j}_\\infty\\leq\\frac{c_1 }{n}+c_2\\tilde{\\delta}^n\\,,\n\t\t\t\\end{equation*}\n\t\t\tfor some constants $c_1,c_2\\ge 0$ depending on all involved parameters except from $n$.\n\t\t\\end{thm}\n\t\t\\subsection{Proof of \\texorpdfstring{\\Cref{thm:spectral-gap-uniform} }{Theorem 3.3 }}~\\\\\n\t\tSimilarly to the papers \\cite{Mobus.2019} and \\cite{Becker.2021}, we use the holomorphic functional calculus to\n\t\tseparate the spectrum of the contraction $Me^{t\\cL}$ appearing in the Zeno sequence. In contrast to \\cite{Becker.2021} where the $C_0$-semigroup is approximated by a sequence of uniformly continuous semigroups, we instead crucially rely upon the uniform continuity of the perturbed contraction to recover the optimal convergence rate. We upper bound the following terms: \n\t\t\\begin{align}\n\t\t\t\\norm{\\left(Me^{\\frac{t}{n}\\cL}\\right)^n-\\sum_{j=1}^{J}\\lambda_j^ne^{tP_j\\cL P_j}P_j}_\\infty&\\leq\\norm{\\left(Me^{\\frac{t}{n}\\cL}\\right)^n-\\left(P_\\Sigma Me^{\\frac{t}{n}\\cL}P_\\Sigma\\right)^n}_\\infty\\label{eq:proofthm2-term1}\\\\\n\t\t\t&+\\norm{\\left(P_\\Sigma Me^{\\frac{t}{n}\\cL}P_\\Sigma\\right)^n-\\sum_{j=1}^{J}\\lambda_j^ne^{n\\left(C_{j}(\\text{\\tiny{$\\tfrac{t}{n}$}})-P_{j}(\\text{\\tiny{$\\tfrac{t}{n}$}})\\right)}P_{j}(\\tfrac{t}{n})}_\\infty\\label{eq:proofthm2-term2}\\\\\n\t\t\t&+\\norm{\\sum_{j=1}^{J}\\lambda_j^ne^{n\\left(C_{j}(\\text{\\tiny{$\\tfrac{t}{n}$}})-P_{j}(\\text{\\tiny{$\\tfrac{t}{n}$}})\\right)}P_{j}(\\tfrac{t}{n})-\\sum_{j=1}^{J}\\lambda_j^ne^{tP_j\\cL P_j}P_j}_\\infty\\label{eq:proofthm2-term3}\n\t\t\\end{align}\n\t\twhere the definitions of the \\textit{perturbed spectral projection} $P_j(\\tfrac{t}{n})$ and the \\textit{Chernoff contraction} $C_{j}(\\tfrac{t}{n})$ are postponed to \\Cref{lem:proofthm2-term2}.\n\t\t\n\t\t\\subsubsection*{Approximation of the Perturbed Spectral Projection:}\n\t\t\t In the following result, we consider an operator $A$ uniformly perturbed by a vector-valued map $t\\mapsto B(t)$ in the following way:\n\t\t\t \\begin{equation*}\n\t\t\t t\\mapsto A+tB(t).\n\t\t\t \\end{equation*}\n\t\t\t Under certain assumptions on the perturbation controlled by $t$, we construct the associated perturbed spectral projection for which we obtain an approximation bound (cf.~\\cite[Lem.~5.3]{Becker.2021}). The key tools are the holomorphic functional calculus and the semicontinuity of the spectrum under \\textit{uniform perturbations}. The statements are summarized in \\Cref{prop:holomorphic-functional-calculus} and \\Cref{lem:semicontinuity-spectrum}.\n\t\t\t\\begin{lem}\\label{lem:quantitative-appro-riesz-projection}\n\t\t\t\tLet $A\\in\\cB(\\cX)$, $t\\mapsto B(t)$ be a vector-valued map on $\\cB(\\cX)$ which is uniformly continuous at $t=0$ with $\\sup_{t\\geq0}\\|B(t)\\|_\\infty\\leq b$, and $\\Gamma:[0,2\\pi]\\rightarrow\\rho(A)$ be a curve separating $\\sigma(A)$. Then, there exists an $\\epsilon>0$ so that for all $t\\in[0,\\epsilon]$\n\t\t\t\t\\begin{equation*}\n\t\t\t\t\tP(t)\\coloneqq \\frac{1}{2\\pi i} \\oint_{\\Gamma}R(z,A+tB(t))dz\n\t\t\t\t\\end{equation*}\n\t\t\t\tdefines a projection with $\\norm{P(t)}_\\infty\\leq\\tfrac{d|\\Gamma|}{2\\pi}$ and derivative at $t=0$ given by\n\t\t\t\t\\begin{equation}\\label{eq:thm2-term1-derivative}\n\t\t\t\t\tP'\\coloneqq\\lim\\limits_{t\\rightarrow0}\\frac{P(t)-{P(0)}}{t}=\\frac{1}{2\\pi i}\\oint_{\\Gamma}R(z,A)B(0)R(z,A)dz.\n\t\t\t\t\\end{equation}\n\t\t\t\twith $\\|P'\\|_\\infty\\leq\\frac{R^2b|\\Gamma|}{2\\pi}$. The zeroth order approximation of $t\\mapsto P(t)$ can be controlled by\n\t\t\t\t\\begin{equation}\\label{eq:zeroth-order-approximation}\n\t\t\t\t \\|P(t)-P\\|_\\infty\\leq \\frac{tRbd|\\Gamma|}{2\\pi}\n\t\t\t\t\\end{equation}\n\t\t\t\tand the first order approximation by \n\t\t\t\t\\begin{equation}\\label{eq:first-order-approximation}\n\t\t\t\t\t\\norm{{P(t)-P-tP'}}_\\infty\\leq\\frac{tR^2|\\Gamma|}{2\\pi}\\left(tb^2d+\\norm{B(t)-B(0)}_\\infty\\right).\n\t\t\t\t\\end{equation}\n\t\t\t\tAbove $|\\Gamma|$ denotes the length of the curve $\\Gamma$, $P$ abbreviates the unperturbed spectral projection $P(0)$, $R\\coloneqq \\sup_{z\\in\\Gamma}\\|R(z,A)\\|_\\infty<\\infty$, and $d=R\\inf_{z\\in\\Gamma}\\frac{2+2|z|^2}{1+2|z|^2}$.\n\t\t\t\\end{lem}\n\t\t\\begin{proof}\n\t\t\tSince $t\\mapsto B(t)$ is uniformly continuous at $t=0$, the vector-valued map $t\\mapsto A+tB(t)$ is uniformly continuous as well. Then, \\Cref{lem:semicontinuity-spectrum} states that there exists an $\\epsilon>0$ such that $\\sigma(A+tB(t))$ is separated by $\\Gamma$ for all $t\\in[0,\\epsilon]$ and \\Cref{prop:holomorphic-functional-calculus} shows that \n\t\t\t\\begin{equation*}\n\t\t\t\tP(t)=\\frac{1}{2\\pi i}\\oint_{\\Gamma}R(z,A+tB(t))dz\n\t\t\t\\end{equation*}\n\t\t\tdefines a projection on $\\cX$ for all $t\\in[0,\\epsilon]$. Let $R\\coloneqq \\sup_{z\\in\\Gamma}\\|R(z,A)\\|_\\infty<\\infty$, then using the same steps as in \\Cref{eq:proofthm1-perturbed-resolvent-uniform-boundedness}, we have that for all $\\eta\\in\\Gamma$\n\t\t\t\\begin{equation}\\label{eq:proofthm2-perturbed-resolvent-uniform-boundedness}\n\t\t\t\t\\norm{R(\\eta,A+tB(t))}_\\infty\\leq R\\inf_{z\\in\\Gamma}\\frac{2+2|z|^2}{1+2|z|^2}\\eqqcolon d\\,.\n\t\t\t\\end{equation}\n\t Therefore, the perturbed resolvent is uniformly bounded. To prove the explicit representation of the derivative and the quantitative approximation, we follow the ideas of \\cite[Lem.~5.2-5.3]{Becker.2021}: \n\t\t\t\\begin{align*}\n\t\t\t\t\\frac{P(t)-P(0)}{t}&=\\frac{1}{t2\\pi i}\\left(\\oint_{\\Gamma}R(z,A+tB(t))dz-\\oint_{\\Gamma}R(z,A)dz\\right)\\\\\n\t\t\t\t&=\\frac{1}{2\\pi i}\\oint_{\\Gamma}R(z,A+tB(t))B(t)R(z,A)dz,\n\t\t\t\\end{align*}\n\t\t\twhich uses the \\textit{second resolvent identity}, i.e.~$R(z,A+tB(t))tB(t)R(z,A)=R(z,A)-R(z,A+tB(t))$ for all $z\\in\\Gamma$ and $t\\in[0,\\epsilon]$ and proves \\Cref{eq:thm2-term1-derivative} by Lebesgue's dominated convergence theorem \\cite[Thm.~3.7.9]{Hille.2000}. Moreover, the above equation proves \\Cref{eq:zeroth-order-approximation}. Finally,\n\t\t\t\\begin{equation}\\label{eq:prooflemapprox-constant}\n\t\t\t\t\\begin{aligned}\n\t\t\t\t\t\\norm{P(t)-P-tP'}_\\infty\n\t\t\t\t\t&\\leq \\frac{t}{2\\pi}\\norm{\\oint_{\\Gamma}(R(z,A+tB(t))-R(z,A))B(t)R(z,A)dz}_\\infty\\\\\n\t\t\t\t\t&\\quad+\\frac{t}{2\\pi}\\norm{\\oint_{\\Gamma}R(z,A)\\left(B(t)-B(0))\\right)R(z,A)dz}_\\infty\\\\\n\t\t\t\t\t&\\leq \\frac{t}{2\\pi}\\norm{\\oint_{\\Gamma}R(z,A+tB(t))tB(t)R(z,A)B(t)R(z,A)dz}_\\infty\\\\\n\t\t\t\t\t&\\quad+\\frac{t}{2\\pi}\\norm{\\oint_{\\Gamma}R(z,A)\\left(B(t)-B(0))\\right)R(z,A)dz}_\\infty\\\\\n\t\t\t\t\t&\\leq \\frac{tR^2|\\Gamma|}{2\\pi}\\left(tb^2d+\\norm{B(t)-B(0)}_\\infty\\right)\\\\\n\t\t\t\t\n\t\t\t\t\\end{aligned}\n\t\t\t\\end{equation}\n\t\t\twhere $|\\Gamma|$ denotes the length of the curve $\\Gamma$.\n\t\t\\end{proof}\n\t\tNow, we are ready to prove \\Cref{thm:spectral-gap-uniform}.\n\t\t\n\t\t\\subsubsection*{Upper bound on \\Cref{eq:proofthm2-term1}:}\n\t\t\\begin{lem}\\label{lem:proofthm2-term1} \n\t\t\tLet $(\\cL,\\cD(\\cL))$ be the generator of a $C_0$-contraction semigroup on $\\cX$ and $M\\in\\cB(\\cX)$ a contraction with the same assumption as in \\Cref{thm:spectral-gap-uniform} and $c_p\\coloneqq\\|\\1-P_\\Sigma\\|_\\infty$. Then, there is an $\\epsilon_1>0$ and $c_2\\geq0$ so that for all $t\\geq0$ and $n\\in\\N$ satisfying $t\\in[0,n\\epsilon_1]$\n\t\t\t\\begin{equation*}\n\t\t\t\t\\norm{\\left(Me^{\\frac{t}{n}\\cL}\\right)^n-\\left(P_\\Sigma Me^{\\frac{t}{n}\\cL}P_\\Sigma\\right)^n}\\leq c_2\\tilde{\\delta}^n+\\frac{tb}{n}+\\frac{1}{n}\\frac{tbc_{p}c_2(2+tbc_{p}c_2)(\\tilde{\\delta}-\\tilde{\\delta}^{n})}{1-\\tilde{\\delta}}e^{2tbc_{p}c_2}\\,.\n\t\t\t\\end{equation*}\n\t\t\\end{lem}\n\n\t\t\\begin{proof}\n\t\t\tAs in the proof of \\Cref{prop:spectral-gap-uniform-norm-power-convergence}, \\Cref{prop:equivalence-spectral-gap} shows that the uniform power conver-\\linebreak\n\t\t\t\\begin{minipage}[c]{0.69\\textwidth}\n\t\t\t\tgence assumption (\\ref{unifpower}), that is $\\|M^n-\\sum_{j=1}^J\\lambda_j^{{ n}}P_j\\|_\\infty\\leq\\tilde{c}\\,\\delta^n$ for all $n\\in\\N$, is equivalent to the spectral gap assumption (q.v. \\Caref{sec:appendix-spectral-gap}),\n\t\t\t\t\\begin{equation*}\n\t\t\t\t\t\\sigma(M)\\subset\\D_\\delta\\cup\\{\\lambda_1,...,\\lambda_J\\},\n\t\t\t\t\\end{equation*}\n\t\t\t\twith corresponding quasinilpotent operators being zero. Therefore, the curve $\\gamma:[0,2\\pi]\\rightarrow\\C,\\varphi\\mapsto\\tilde{\\delta}e^{i\\varphi}$, with $\\tilde{\\delta}\\in (\\delta,1)$, encloses the spectrum of $M^\\perp\\coloneqq MP_\\Sigma^\\perp=P_\\Sigma^\\perp M$, where $P_\\Sigma=\\sum_{j=1}^JP_j$ and $P_\\Sigma^\\perp=\\1-P_\\Sigma$ (q.v.~\\Caref{fig1}). By \\Cref{lem:properties-semigroups}\n\t\t\t\t\\begin{equation*}\n\t\t\t\t\t\\norm{M^\\perp e^{s\\cL}-M^\\perp}_\\infty=s\\norm{M^\\perp\\cL\\int_{0}^{1}e^{\\tau s\\cL}d\\tau}_\\infty\\leq sc_{p}b\n\t\t\t\t\\end{equation*}\n\t\t\t with $c_{p}\\coloneqq\\|P_{\\Sigma}^\\perp\\|_\\infty$. Therefore, $M^\\perp e^{s\\cL}$ converges uniformly to $M^\\perp$ for $s\\downarrow0$. Hence, \\Cref{lem:semicontinuity-spectrum} shows that there exists an \\linebreak\\vspace{-2ex}\n\t\t\t\\end{minipage}\n\t\t\t\\begin{minipage}[c]{0.3\\textwidth}\n\t\t\t\t\\begin{center}\n\t\t\t\t\t\\begin{tikzpicture}\n\t\t\t\t\t\t\\draw[dashed,tumivory] (0,0) ellipse (1.4cm and 1.4cm);\n\t\t\t\t\t\t\\draw[dashed,mygreen] (0,0) ellipse (1.15cm and 1.15cm);\n\t\t\t\t\t\t\\filldraw (0,0) ellipse (0.03cm and 0.03cm);\n\t\t\t\t\t\t\n\t\t\t\t\t\t\\filldraw (15:1.4) ellipse (0.03cm and 0.03cm);\n\t\t\t\t\t\n\t\t\t\t\t\t\\draw (15:1.7) node {$\\lambda_1$};\n\t\t\t\t\t\t\n\t\t\t\t\t\t\\filldraw (70:1.4) ellipse (0.03cm and 0.03cm);\n\t\t\t\t\t\n\t\t\t\t\t\t\\draw (70:1.7) node {$\\lambda_2$};\n\t\t\t\t\t\t\n\t\t\t\t\t\t\\filldraw (110:1.4) ellipse (0.03cm and 0.03cm);\n\t\t\t\t\t\n\t\t\t\t\t\t\\draw (110:1.7) node {$\\lambda_3$};\n\t\t\t\t\t\n\t\t\t\t\t\t\\filldraw (130:1.4) ellipse (0.03cm and 0.03cm);\n\t\t\t\t\t\n\t\t\t\t\t\t\\draw (130:1.7) node {$\\lambda_4$};\n\t\t\t\t\t\t\n\t\t\t\t\t\t\\filldraw (170:1.4) ellipse (0.03cm and 0.03cm);\n\t\t\t\t\t\n\t\t\t\t\t\t\\draw (170:1.7) node {$\\lambda_5$};\n\t\t\t\t\t\t\n\t\t\t\t\t\t\\filldraw (195:1.4) ellipse (0.03cm and 0.03cm);\n\t\t\t\t\t\n\t\t\t\t\t\t\\draw (195:1.7) node {$\\lambda_6$};\n\t\t\t\t\t\t\n\t\t\t\t\t\t\\filldraw (260:1.4) ellipse (0.03cm and 0.03cm);\n\t\t\t\t\t\n\t\t\t\t\t\t\\draw (260:1.7) node {$\\lambda_7$};\n\t\t\t\t\t\t\n\t\t\t\t\t\t\\filldraw (300:1.4) ellipse (0.03cm and 0.03cm);\n\t\t\t\t\t\n\t\t\t\t\t\t\\draw (300:1.7) node {$\\lambda_8$};\n\t\t\t\t\t\t\n\t\t\t\t\t\t\\filldraw (350:1.4) ellipse (0.03cm and 0.03cm);\n\t\t\t\t\t\n\t\t\t\t\t\t\\draw (350:1.7) node {$\\lambda_J$};\n\t\t\t\t\t\t\n\t\t\t\t\t\t\\draw [dotted,domain=310:340] plot ({1.7*cos(\\x)}, {1.7*sin(\\x)});\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t\t\\draw[<->] (0.0,0.05)--(0,0.95);\n\t\t\t\t\t\t\\draw[<->,mygreen] (0.0433,-0.025)--(0.9526cm,-0.55cm);\n\t\t\t\t\t\t\n\t\t\t\t\t\t\\fill[pattern=my north east lines] (0,0) ellipse (1cm and 1cm);\n\t\t\t\t\t\t\\filldraw[white] (-0.35,0.5) ellipse (0.15cm and 0.25cm);\n\t\t\t\t\t\t\\draw (-0.35,0.5) node {$\\delta$};\n\t\t\t\t\t\t\\filldraw[white] (0.35,-0.5) ellipse (0.15cm and 0.25cm);\n\t\t\t\t\t\t\\draw[mygreen] (0.35,-0.5) node {$\\tilde{\\delta}$};\n\t\t\t\t\t\t\\filldraw[white] (-1.06,-1.06) ellipse (0.15cm and 0.3cm);\n\t\t\t\t\t\t\\draw (-1.06,-1.06) node [mygreen]{$\\gamma$};\n\t\t\t\t\t\t\n\t\t\t\t\t\t\\draw (1.5,1.2) node {$\\sigma(M)$};\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t\\end{tikzpicture}\n\t\t\t\t\t\\captionof{figure}{}\\label{fig1}\n\t\t\t\t\\end{center}\n\t\t\t\\end{minipage}\\\\\n\t\t\t$\\epsilon_1>0$ such that the spectrum of $M^\\perp e^{s\\cL}$ can be separated by $\\gamma$ for all $s\\in[0,\\epsilon_1]$. Therefore, we can apply the holomorphic functional calculus (\\Caref{prop:holomorphic-functional-calculus}) to conclude that for all $t\\in[0,n\\epsilon_1]$\n\t\t\t\\begin{equation*}\n\t\t\t\t\\left(M^\\perp e^{\\frac{t}{n}\\cL}\\right)^{k}=\\frac{1}{2\\pi i}\\oint_{\\gamma}z^{k}R(z,M^\\perp e^{\\frac{t}{n}})dz,\n\t\t\t\\end{equation*}\n\t\t\twhere $k\\in\\{1,..,n\\}$. By \\Cref{eq:proofthm1-perturbed-resolvent-uniform-boundedness} and with $c_2\\coloneqq\\sup_{z\\in\\gamma}\\|R(z,M^\\perp)\\|_\\infty \\frac{2+2\\tilde{\\delta}^2}{1+2\\tilde{\\delta}^2}$,\n\t\t\t\\begin{equation*}\n\t\t\t\t\\norm{\\left(M^\\perp e^{\\frac{t}{n}\\cL}\\right)^{k-1}}_\\infty\\leq\\frac{1}{2\\pi}\\oint_{\\gamma}\\abs{z}^{k-1}\\norm{R(z,M^\\perp e^{\\frac{t}{n}})}_\\infty dz\\leq c_2\\tilde{\\delta}^k.\n\t\t\t\\end{equation*}\n\t\t\tMoreover, by the assumptions $\\|P_\\Sigma\\|_\\infty=1$, $\\|M\\cL\\|_\\infty\\leq b$, $\\|\\cL P_\\Sigma\\|_\\infty\\leq b$, and \\Cref{lem:properties-semigroups}\n\t\t\t\\begin{equation*}\n\t\t\t \\norm{MP_\\Sigma e^{t\\cL}P_\\Sigma^\\perp M}_\\infty\\leq tb\\quad\\text{and}\\quad \\norm{MP_\\Sigma^\\perp e^{t\\cL}P_\\Sigma M}_\\infty\\leq tc_{p}b.\n\t\t\t\\end{equation*}\n\t\t\tBy the same expansion of $(Me^{\\frac{t}{n}\\cL})^n=(P_\\Sigma Me^{\\frac{t}{n}\\cL}+M^\\perp e^{\\frac{t}{n}\\cL})^n$ as in the proof of \\Cref{prop:spectral-gap-uniform-norm-power-convergence},\n\t\t\t\\begin{equation*}\n\t\t\t\t\\begin{aligned}\n\t\t\t\t\t\\norm{\\left(Me^{\\frac{t}{n}\\cL}\\right)^n-\\left(P_\\Sigma Me^{\\frac{t}{n}\\cL}\\right)^n}_\\infty\\leq c_2\\tilde{\\delta}^n+\\frac{1}{n}\\frac{tbc_{p}c_2(2+tbc_{p}c_2)(\\tilde{\\delta}-\\tilde{\\delta}^{n})}{1-\\tilde{\\delta}}e^{2tbc_{p}c_2}.\n\t\t\t\t\\end{aligned}\n\t\t\t\\end{equation*}\n\t\t\tFinally, \\Cref{lem:properties-semigroups} shows \n\t\t\t\\begin{equation*}\n\t\t\t\t\\norm{P_\\Sigma Me^{\\frac{t}{n}\\cL}(\\1-P_\\Sigma)}_\\infty=\\frac{t}{n}\\norm{P_\\Sigma M\\cL\\int_{0}^{1} e^{\\tau\\frac{t}{n}\\cL}(\\1-P_\\Sigma)d\\tau}_\\infty\\leq\\frac{tbc_{p}}{n}\n\t\t\t\\end{equation*}\n\t\t\tso that\n\t\t\t\\begin{equation*}\n\t\t\t\t\\norm{\\left(Me^{\\frac{t}{n}\\cL}\\right)^n-\\left(P_\\Sigma Me^{\\frac{t}{n}\\cL}P_\\Sigma\\right)^n}\\leq c_2\\tilde{\\delta}^n+\\frac{tb}{n}+\\frac{1}{n}\\frac{tbc_{p}c_2(2+tbc_{p}c_2)(\\tilde{\\delta}-\\tilde{\\delta}^{n})}{1-\\tilde{\\delta}}e^{2tbc_{p}c_2},\n\t\t\t\\end{equation*}\n\t\t\twhich finishes the proof.\n\t\t\\end{proof}\n\t\t\n\t\t\n\t\\subsubsection*{Upper bound on \\texorpdfstring{\\Cref{eq:proofthm2-term2}}{Equation (2.1)}}\\label{subsec:proofthm2-term2}~\\\\\n\t\tAs in \\Cref{lem:proofthm1-term2}, we apply the modified Chernoff Lemma (\\Caref{lem:improved-chernoff}) to upper bound the second term (\\ref{eq:proofthm2-term2}). \n\t\tHowever, our proof strategy includes two crucial improvements compared to Theorem 3 in \\cite{Becker.2021}. Firstly, we show that the spectrum of the perturbed contraction is upper semicontinuous under certain assumptions on $M$ and the $C_0$-semigroup. Therefore, we can use the holomorphic functional calculus and apply the modified Chernoff Lemma with respect to each eigenvalue separately, which allows us to achieve the optimal convergence as in \\Cref{lem:proofthm1-term2}. \n\t\t\\begin{lem}\\label{lem:proofthm2-term2}\n\t\t\tLet $(\\cL,\\cD(\\cL))$ be the generator of a $C_0$-contraction semigroup on $\\cX$ and $M\\in\\cB(\\cX)$ a contraction satisfying the same assumption as in \\Cref{thm:spectral-gap-uniform}. Then, there is an $\\epsilon_2>0$, and a $\\tilde{d}_1\\geq0$ so that for all $t\\geq0$ and $n\\in\\N$ satisfying $t\\in[0,n\\epsilon_2]$ \n\t\t\t\\begin{equation*}\n\t\t\t\t\\norm{\\left(P_\\Sigma Me^{\\frac{t}{n}\\cL}P_\\Sigma\\right)^n-\\sum_{j=1}^{J}\\lambda_j^ne^{n\\left(C_{j}(\\text{\\tiny{$\\tfrac{t}{n}$}})-P_{j}(\\text{\\tiny{$\\tfrac{t}{n}$}})\\right)}P_{j}(\\tfrac{t}{n})}_\\infty\\leq\\frac{J}{n}e^{t\\tilde{d}_1}\n\t\t\t\\end{equation*}\n\t\twhere $C_{j}(\\tfrac{1}{n})\\coloneqq\\bar{\\lambda}_jP_{j}(\\tfrac{1}{n})P_\\Sigma Me^{\\frac{1}{n}\\cL}P_\\Sigma P_{j}(\\tfrac{1}{n})$ and $P_\\Sigma\\coloneqq\\sum_{j=1}^{J}P_j$.\n\t\t\\end{lem}\n\t\t\\begin{proof}\n\t\t\tBy \\Cref{prop:equivalence-spectral-gap}, the uniform power convergence (\\ref{unifpower}) shows that the $P_j$'s are the eigenprojections of $M$ so that $P_\\Sigma M=MP_\\Sigma=\\sum_{j=1}^{J}\\lambda_jP_j$ and the spectrum $\\sigma(P_\\Sigma M)$ consists of\\linebreak\n\t\t\t\\begin{minipage}[c]{0.69\\textwidth}\n\t\t\t $J$ isolated eigenvalues on the unit circle separated by the curves $\\Gamma_j:[0,2\\pi]\\rightarrow\\C,\\phi\\mapsto\\lambda_j+re^{i\\phi}$ (q.v.~\\Caref{sec:prelim} and \\Caref{fig2}) with radius\n\t\t\t\t\\begin{equation}\\label{eq:defr}\n\t\t\t\t\tr\\coloneqq\\min_{i\\neq j}\\left\\{\\frac{\\abs{\\lambda_i-\\lambda_j}}{3}\\right\\}.\n\t\t\t\t\\end{equation}\n\t\t\t\tNote that we use the curve interchangeably with its image and denote the formal sum of all curves around the eigenvalues $\\{\\lambda_j\\}_{j=1}^J$ by $\\Gamma$. In the following, we define the vector-valued function:\n\t\t\t\t\\begin{equation}\\label{eq:defB(t)}\n\t\t\t\t\t\\begin{aligned}\n\t\t\t\t\t\ts\\mapsto P_\\Sigma M+sB(s)&\\coloneqq P_\\Sigma M+sP_\\Sigma M\\cL\\int_0^1e^{s\\tau \\cL}P_\\Sigma d\\tau\\\\\n\t\t\t\t\t\t&\\;=P_\\Sigma Me^{s\\cL}P_\\Sigma\\,.\n\t\t\t\t\t\\end{aligned}\n\t\t\t\t\\end{equation}\n\t\t\t\\end{minipage}\n\t\t\t\\begin{minipage}[c]{0.3\\textwidth}\n\t\t\t\t\\begin{center}\n\t\t\t\t\t\\begin{tikzpicture}\n\t\t\t\t\t\t\\draw[dashed,tumivory] (0,0) ellipse (1.4cm and 1.4cm);\n\t\t\t\t\t\t\\draw[dashed,mygreen] (0,0) ellipse (1.15cm and 1.15cm);\n\t\t\t\t\t\t\\filldraw (0,0) ellipse (0.03cm and 0.03cm);\n\t\t\t\t\t\t\n\t\t\t\t\t\t\\filldraw (15:1.4) ellipse (0.03cm and 0.03cm);\n\t\t\t\t\t\t\\draw[dashed,myorange] (15:1.4) ellipse (0.2cm and 0.2cm);\n\t\t\t\t\t\t\\draw[myorange] (15:1.9) node {$\\Gamma_1$};\n\t\t\t\t\t\t\n\t\t\t\t\t\t\\filldraw (70:1.4) ellipse (0.03cm and 0.03cm);\n\t\t\t\t\t\t\\draw[dashed,myorange] (70:1.4) ellipse (0.2cm and 0.2cm);\n\t\t\t\t\t\t\\draw[myorange] (70:1.9) node {$\\Gamma_2$};\n\t\t\t\t\t\t\n\t\t\t\t\t\t\\filldraw (110:1.4) ellipse (0.03cm and 0.03cm);\n\t\t\t\t\t\t\\draw[dashed,myorange] (110:1.4) ellipse (0.2cm and 0.2cm);\n\t\t\t\t\t\t\\draw[myorange] (110:1.9) node {$\\Gamma_3$};\n\t\t\t\t\t\t\n\t\t\t\t\t\t\\filldraw (130:1.4) ellipse (0.03cm and 0.03cm);\n\t\t\t\t\t\t\\draw[dashed,myorange] (130:1.4) ellipse (0.2cm and 0.2cm);\n\t\t\t\t\t\t\\draw[myorange] (130:1.9) node {$\\Gamma_4$};\n\t\t\t\t\t\t\n\t\t\t\t\t\t\\filldraw (170:1.4) ellipse (0.03cm and 0.03cm);\n\t\t\t\t\t\t\\draw[dashed,myorange] (170:1.4) ellipse (0.2cm and 0.2cm);\n\t\t\t\t\t\t\\draw[myorange] (170:1.9) node {$\\Gamma_5$};\n\t\t\t\t\t\t\n\t\t\t\t\t\t\\filldraw (195:1.4) ellipse (0.03cm and 0.03cm);\n\t\t\t\t\t\t\\draw[dashed,myorange] (195:1.4) ellipse (0.2cm and 0.2cm);\n\t\t\t\t\t\t\\draw[myorange] (195:1.9) node {$\\Gamma_6$};\n\t\t\t\t\t\t\n\t\t\t\t\t\t\\filldraw (260:1.4) ellipse (0.03cm and 0.03cm);\n\t\t\t\t\t\t\\draw[dashed,myorange] (260:1.4) ellipse (0.2cm and 0.2cm);\n\t\t\t\t\t\t\\draw[myorange] (260:1.9) node {$\\Gamma_7$};\n\t\t\t\t\t\t\n\t\t\t\t\t\t\\filldraw (300:1.4) ellipse (0.03cm and 0.03cm);\n\t\t\t\t\t\t\\draw[dashed,myorange] (300:1.4) ellipse (0.2cm and 0.2cm);\n\t\t\t\t\t\t\\draw[myorange] (300:1.9) node {$\\Gamma_8$};\n\t\t\t\t\t\t\n\t\t\t\t\t\t\\filldraw (350:1.4) ellipse (0.03cm and 0.03cm);\n\t\t\t\t\t\t\\draw[dashed,myorange] (350:1.4) ellipse (0.2cm and 0.2cm);\n\t\t\t\t\t\t\\draw[myorange] (350:1.9) node {$\\Gamma_J$};\n\t\t\t\t\t\t\n\t\t\t\t\t\t\\draw [dotted,myorange,domain=310:340] plot ({1.9*cos(\\x)}, {1.9*sin(\\x)});\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t\t\\draw[<->] (0.0,0.05)--(0,0.95);\n\t\t\t\t\t\t\\draw[<->,mygreen] (0.0433,-0.025)--(0.9526cm,-0.55cm);\n\t\t\t\t\t\t\n\t\t\t\t\t\t\\fill[pattern=my north east lines] (0,0) ellipse (1cm and 1cm);\n\t\t\t\t\t\t\\filldraw[white] (-0.35,0.5) ellipse (0.15cm and 0.25cm);\n\t\t\t\t\t\t\\draw (-0.35,0.5) node {$\\delta$};\n\t\t\t\t\t\t\\filldraw[white] (0.35,-0.5) ellipse (0.15cm and 0.25cm);\n\t\t\t\t\t\t\\draw[mygreen] (0.35,-0.5) node {$\\tilde{\\delta}$};\n\t\t\t\t\t\t\\filldraw[white] (-1.06,-1.06) ellipse (0.15cm and 0.3cm);\n\t\t\t\t\t\t\\draw (-1.06,-1.06) node [mygreen]{$\\gamma$};\n\t\t\t\t\t\t\n\t\t\t\t\t\t\\draw (1.5,1.2) node {$\\sigma(M)$};\n\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t\\end{tikzpicture}\n\t\t\t\t\t\\captionof{figure}{}\\label{fig2}\n\t\t\t\t\\end{center}\n\t\t\t\\end{minipage}\\\\[1ex]\n\t\t\tSince $M\\cL$ is bounded, the defined vector-valued map converges in the uniform topology to $P_\\Sigma M$. Moreover, \\Cref{lem:properties-semigroups} shows that $s\\mapsto B(s)$ is uniformly bounded and continuous in $s=0$ because\n\t\t\t\\begin{equation}\\label{eq:proofthm2-continuity-b}\n\t\t\t\t\\norm{B(s)-B(0)}_\\infty=s\\norm{P_\\Sigma M\\cL\\int_{0}^{1}\\int_0^{1}\\tau_1e^{\\tau_1\\tau_2s\\cL}\\cL P_\\Sigma d\\tau_2d\\tau_1}_\\infty\\leq s\\frac{b^2}{2}\n\t\t\t\\end{equation}\n\t\t\twhere we have used the assumption $\\|P_\\Sigma\\|_\\infty\\leq 1$. Then, we can apply \\Cref{lem:quantitative-appro-riesz-projection} which shows that there exists an $\\epsilon_2>0$ such that for all $s\\in[0,\\epsilon_2]$\n\t\t\t\\begin{equation*}\n\t\t\t\tP_j(s)\\coloneqq\\frac{1}{2\\pi i}\\oint_{\\Gamma_j}R(z,P_\\Sigma M+sB(s))dz\n\t\t\t\\end{equation*}\n\t\t\tdefines the perturbed spectral projection w.r.t.~$\\lambda_j$. Next, let $t\\geq0$, $n\\in\\N$ such that $t\\in[0,n\\epsilon_2]$. By \\Cref{lem:quantitative-appro-riesz-projection} and \\Cref{eq:proofthm2-continuity-b}, the perturbed spectral projection can be approximated by\n\t\t\t\\begin{align}\n\t\t\t \\norm{P_j(\\tfrac{t}{n})-P_j}_\\infty&\\leq\\frac{t}{n}R_jb\\left(d_j-\\frac{1}{2}\\right)r\\leq\\frac{t}{n}R_jbd_jr\\eqqcolon\\frac{t}{n}v_j \\label{eq:profthm2term2-approx-constant0}\\\\ \n\t\t\t \\norm{P_j(\\tfrac{t}{n})-P_j-\\tfrac{t}{n}P_j'}_\\infty&\\leq\\frac{t^2}{n^2}R_j^2b^2rd_j\\label{eq:profthm2term2-approx-constant}\n\t\t\t\\end{align}\n\t\t\twhere $R_j\\coloneqq\\sup_{z\\in\\Gamma_j}\\|R(z,P_\\Sigma M)\\|_\\infty$, $d_j\\coloneqq R_j\\inf_{z\\in\\Gamma_j}\\frac{2+2|z|^2}{1+2|z|^2}+\\frac{1}{2}$, and we use that $|\\Gamma_j|=2\\pi r$. Note that the defined $d_j$ is not exactly the $d$ in \\Cref{lem:quantitative-appro-riesz-projection}. Moreover, note that $\\|P_j(\\tfrac{t}{n})\\|_\\infty\\leq d_jr$ and $\\|P'_j\\|_\\infty\\leq R_j^2br$. By the spectral decomposition,\n\t\t\t\\begin{equation}\\label{eq:proof-thm2-spectral-decomp}\n\t\t\t\t\\left(P_\\Sigma Me^{\\frac{t}{n}\\cL}P_\\Sigma \\right)^n=\\sum_{j=1}^{J}\\left(P_j\\left(\\tfrac{t}{n}\\right)P_\\Sigma Me^{\\frac{t}{n}\\cL}P_\\Sigma P_j\\left(\\tfrac{t}{n}\\right)\\right)^n.\n\t\t\t\\end{equation}\n\t\t\tNext, we aim at applying the modified Chernoff lemma \\ref{lem:improved-chernoff} to $C_{j}(\\tfrac{t}{n})\\coloneqq\\bar{\\lambda}_jP_{j}(\\tfrac{t}{n})P_\\Sigma Me^{\\frac{1}{n}\\cL}P_\\Sigma P_{j}(\\tfrac{1}{n})$ for all $j\\in\\{1,..,J\\}$, which has to be adapted since it is no longer clear that $\\|C_j(\\tfrac{1}{n})\\|_\\infty=1$. We start by bounding the difference in norm between $C_t(\\tfrac{t}{n})$ and $P_j(\\tfrac{t}{n})$. By the fundamental theorem of calculus and the facts $\\|P_j(\\tfrac{t}{n})\\|_\\infty\\leq d_jr$, $\\|P_{\\Sigma}M\\cL\\|_\\infty\\leq b$, $\\|e^{s\\cL}\\|_\\infty\\leq 1$, $|\\lambda_j|=1$\n\t\t\t\\begin{equation}\\label{eq:proofthm2term2-approx1}\n\t\t\t\t\\begin{aligned}\n\t\t\t\t\t\\norm{C_{j}(\\tfrac{t}{n})-P_j(\\tfrac{t}{n})}_\\infty&\\leq\\frac{t}{n}\\norm{\\bar{\\lambda}_jP_j(\\tfrac{t}{n})P_\\Sigma M\\cL\\int_{0}^{1}e^{\\frac{st}{n}\\cL}P_\\Sigma P_j(\\tfrac{t}{n})ds}_\\infty\\\\\n\t\t\t\t\t&\\quad+\\norm{P_j(\\tfrac{t}{n})P_\\Sigma MP_j(\\tfrac{t}{n})-P_j(\\tfrac{t}{n})}_\\infty\\\\\n\t\t\t\t\t&\\leq\\frac{t}{n}bd_j^2r^2+\\norm{\\bar{\\lambda}_jP_j(\\tfrac{t}{n})P_\\Sigma MP_j(\\tfrac{t}{n})-P_j(\\tfrac{t}{n})}_\\infty.\n\t\t\t\t\\end{aligned}\n\t\t\t\\end{equation}\n\t\t\tIn the next step, we focus on the second term and prove a higher order approximation then needed because in \\Cref{lem:proofthm2-term3} we will reuse this calculation. In the following calculation, we use the bounds from above, in particular \\Cref{eq:profthm2term2-approx-constant0} and (\\ref{eq:profthm2term2-approx-constant}). Moreover, we use the product rule for derivatives, which shows $P_j'=P_jP_j'+P_j'P_j$ by $\\tfrac{\\partial}{\\partial s}P_j(s)=\\tfrac{\\partial}{\\partial s}P_j(s)^2$ (cf.~\\cite[Lem.~3]{Mobus.2019}).\n\t\t\t\\begin{equation}\\label{eq:proofthm2term2-approx2}\n\t\t\t \\begin{aligned}\n\t\t\t\t\t&\\norm{\\bar{\\lambda}_jP_j(\\tfrac{t}{n})P_\\Sigma MP_j(\\tfrac{t}{n})-P_j(\\tfrac{t}{n})}_\\infty\\\\\n\t\t\t\t\t&\\qquad=\\norm{\\left(P_j(\\tfrac{t}{n})-P_j-\\tfrac{t}{n}P_j'\\right)P_\\Sigma MP_j(\\tfrac{t}{n})}_\\infty+\\norm{P_jP_j(\\tfrac{t}{n})+\\frac{t}{n}P_j'P_\\Sigma MP_j(\\tfrac{t}{n})-P_j(\\tfrac{t}{n})}_\\infty\\\\\n\t\t\t\t\t&\\qquad\\leq\\begin{aligned}[t]\n\t\t\t\t\t \\frac{t^2}{n^2}R_j^2b^2r^2d_j^2&+\\norm{P_j\\left(P_j(\\tfrac{t}{n})-P_j-\\frac{t}{n}P_j'\\right)}_\\infty+\\frac{t}{n}\\norm{P_j'P_\\Sigma M\\left(P_j(\\tfrac{t}{n})-P_j\\right)}_\\infty\\\\\n\t\t\t\t\t &+\\norm{P_j+\\frac{t}{n}\\left(P_jP_j'+P_j'P_j\\right)-P_j(\\tfrac{t}{n})}_\\infty\n\t\t\t\t\t\\end{aligned}\\\\\n\t\t\t\t\t&\\qquad\\leq\\frac{t^2}{n^2}R_j^2b^2r^2d_j^2+2\\frac{t^2}{n^2}R_j^2b^2rd_j+\\frac{t^2}{n^2}R_j^3b^2d_jr^2\\\\\n\t\t\t\t\t&\\qquad\\leq\\frac{t^2}{n^2}R_j^2b^2d_jr\\left(d_jr+R_jr+2\\right)\n\t\t\t\t\\end{aligned}\n\t\t\t\\end{equation}\n\t\t\tCombining \\Cref{eq:proofthm2term2-approx1}, (\\ref{eq:proofthm2term2-approx2}), and $\\tfrac{t}{n}\\leq\\epsilon_2$ shows\n\t\t\t\\begin{equation}\\label{eq:proofthm2term2-approx}\n\t\t\t\t\\begin{aligned}\n\t\t\t\t\t\\norm{C_{j}(\\tfrac{t}{n})-P_j(\\tfrac{t}{n})}_\\infty&\\leq\\frac{t}{n}bd_j^2r^2+\\frac{t^2}{n^2}R_j^2b^2d_jr\\left(d_jr+R_jr+2\\right)\\\\\n\t\t\t\t\t&\\leq\\frac{t}{n}bd_jr\\left(d_jr+\\epsilon_2R_j^2b\\left(d_jr+R_jr+2\\right)\\right)\\eqqcolon\\frac{t}{n}w_j\n\t\t\t\t\\end{aligned}\n\t\t\t\\end{equation}\n\t\t\tIn \\Cref{eq:profthm2term2-approx-constant0} and (\\ref{eq:proof-thm2-chernoff}), we have proven that $\\|P_j(\\tfrac{t}{n})-P_j\\|\\leq\\tfrac{t}{n}v_j$, $\\|C_j(\\tfrac{t}{n})-P_j(\\tfrac{t}{n})\\|\\leq\\tfrac{t}{n}w_j$ and note that $\\|P_j\\|_\\infty=1$, $P_j(\\tfrac{t}{n})C_j(\\tfrac{t}{n})=C_j(\\tfrac{t}{n})P_j(\\tfrac{t}{n})=C_j(\\tfrac{t}{n})$ holds by definition. Then we can apply the approximate version of the modified Chernoff \\Cref{lem:approx-improved-chernoff} to $C_j(\\tfrac{t}{n})$. This shows\n\t\t\t\\begin{equation}\\label{eq:proof-thm2-chernoff}\n\t\t\t \\norm{C_j(\\tfrac{t}{n})^n-e^{n(C_j(\\text{\\tiny{$\\tfrac{t}{n}$}})-P_j(\\text{\\tiny{$\\tfrac{t}{n}$}}))}P_j(\\tfrac{t}{n})}_\\infty\\leq\\frac{t^2w_j^2}{2n}e^{t(v_j+w_j)}\\leq\\frac{1}{n}e^{t(v_j+2w_j)}.\n\t\t\t\\end{equation}\n\t\t\tCombining \\Cref{eq:proof-thm2-spectral-decomp} and (\\ref{eq:proof-thm2-chernoff}) and writing out the constants $v_j$ and $w_j$ gives\n\t\t\t\\begin{align*}\n\t\t\t\t&\\norm{\\left(P_\\Sigma Me^{\\frac{t}{n}\\cL}P_\\Sigma\\right)^n-\\sum_{j=1}^{J}\\lambda_j^ne^{n\\left(C_{j}(\\text{\\tiny{$\\tfrac{t}{n}$}})-P_{j}(\\text{\\tiny{$\\tfrac{t}{n}$}})\\right)}P_{j}(\\tfrac{t}{n})}_\\infty\\\\\n\t\t\t\t&\\hspace{10ex}\\leq J\\max\\limits_{j\\in\\{1,..,J\\}}\\norm{\\lambda_j^n\\left(C_j(\\tfrac{t}{n})\\right)^n-\\lambda_j^ne^{n\\left(C_{j}(\\text{\\tiny{$\\tfrac{t}{n}$}})-P_{j}(\\text{\\tiny{$\\tfrac{t}{n}$}})\\right)}P_{j}(\\tfrac{t}{n})}_\\infty\\\\\n\t\t\t\t&\\hspace{10ex}\\leq \\frac{J}{n}e^{t(v_j+2w_j)}\\\\\n\t\t\t\\end{align*}\n\t\t\tFinally, we define $\\tilde{d}_1=\\max\\limits_{j\\in\\{1,...,J\\}}v_j+2w_j$, which finishes the proof.\n\t\t\\end{proof}\n\t\t\n\t\t\\subsubsection*{Upper bound on \\Cref{eq:proofthm2-term3}:}\n\t\t\\begin{lem}\\label{lem:proofthm2-term3}\n\t\t\tLet $(\\cL,\\cD(\\cL))$ be the generator of a $C_0$-contraction semigroup on $\\cX$ and $M\\in\\cB(\\cX)$ a contraction with the same assumption as in \\Cref{thm:spectral-gap-uniform}. Then, there exists a constant $\\tilde{d}_2\\geq0$ such that \n\t\t\t\\begin{equation*}\n\t\t\t\t\\norm{\\sum_{j=1}^{J}\\lambda_j^ne^{n\\left(C_{j}(\\text{\\tiny{$\\tfrac{t}{n}$}})-P_{j}(\\text{\\tiny{$\\tfrac{t}{n}$}})\\right)}P_{j}(\\tfrac{t}{n})-\\sum_{j=1}^{J}\\lambda_j^ne^{tP_j\\cL P_j}P_j}_\\infty\\leq\\frac{J}{n}e^{t\\tilde{d}_2}\\max\\limits_{s\\in[0,1]}\\norm{e^{stP_j\\cL P_j}}_\\infty.\n\t\t\t\\end{equation*}\n\t\t\\end{lem}\n\t\t\\begin{proof}\n\t\t\tFor ease of notation, we absorb the time parameter $t$ into the generator $\\cL$ and $b$. In order to prove the convergence of the generator, \\Cref{eq:proofthm2term2-approx2} proves:\n\t\t\t\\begin{equation*}\n\t\t\t\t\\begin{aligned}\n\t\t\t\t\t\\norm{n\\left(C_{j}(\\tfrac{1}{n})-P_j(\\tfrac{1}{n})\\right)-P_j\\cL P_j}_\\infty&\\leq\\norm{\\bar{\\lambda}_jP_j(\\tfrac{1}{n})P_\\Sigma M\\cL\\int_{0}^{1}e^{\\frac{s}{n}\\cL}P_\\Sigma P_j(\\tfrac{1}{n})ds-P_j\\cL P_j}_\\infty\\\\\n\t\t\t\t\t&\\quad+\\frac{1}{n}R_j^2b^2d_jr\\left(d_jr+R_jr+2\\right),\n\t\t\t\t\\end{aligned}\n\t\t\t\\end{equation*}\n\t\twhere $R_j\\coloneqq\\sup_{z\\in\\Gamma_j}\\|R(z,P_\\Sigma M)\\|_\\infty$, $d_j\\coloneqq R_j\\inf_{z\\in\\Gamma_j}\\frac{2+2|z|^2}{1+2|z|^2}+\\frac{1}{2}$, and $r$ is the radius of the curves $\\Gamma_j$ defined in \\Cref{eq:defr}. Then, we apply \\Cref{lem:properties-semigroups} and \\Cref{lem:quantitative-appro-riesz-projection} on the first term:\n\t\t\t\\begin{align*}\n\t\t\t\t&\\norm{\\bar{\\lambda}_jP_j(\\tfrac{1}{n})P_\\Sigma M\\cL\\int_{0}^{1}e^{\\frac{\\tau_1}{n}\\cL}P_\\Sigma P_j(\\tfrac{1}{n})d\\tau_1- P_j\\cL P_j}_\\infty\\\\\n\t\t\t &\\qquad \\leq\\frac{1}{n}\\norm{\\bar{\\lambda}_jP_j(\\tfrac{1}{n})P_\\Sigma M\\cL\\int_{0}^{1}\\int_{0}^{1}\\tau_1e^{\\frac{\\tau_1\\tau_2}{n}\\cL}\\cL P_\\Sigma P_j(\\tfrac{1}{n})d\\tau_2d\\tau_1}_\\infty\\\\\t\t&\\qquad\\quad+\\norm{\\bar{\\lambda}_jP_j(\\tfrac{1}{n})P_\\Sigma M\\cL P_\\Sigma P_j(\\tfrac{1}{n})-P_j\\cL P_j}_\\infty\\\\\n\t\t\t &\\qquad\\leq\\frac{1}{2n}b^2d_j^2r^2+\\norm{\\left(P_j(\\tfrac{1}{n})-P_j\\right)P_\\Sigma M\\cL P_\\Sigma P_j(\\tfrac{1}{n})}_\\infty+\\norm{P_j\\cL \\left(P_j(\\tfrac{1}{n})-P_j\\right)}\\\\\n\t\t\t &\\qquad\\leq\\frac{1}{n}b^2d_jr\\left(\\frac{1}{2}d_jr+R_jd_jr+R_j\\right)\\,,\n\t\t\t\\end{align*}\n\t\t\twhere we used \\Cref{eq:profthm2term2-approx-constant0} in the last step and the assumption that $M\\cL$ and $\\cL P_{\\Sigma}$ are bounded by $b$ and all the inequalities discussed before \\Cref{eq:proofthm2term2-approx}. In combination with \\Cref{lem:integral-equation-semigroups} \n\t\t\t\\begin{align*}\n\t\t\t\t&\\norm{e^{n\\left(C_{j}(\\text{\\tiny{$\\tfrac{1}{n}$}})-P_{j}(\\text{\\tiny{$\\tfrac{1}{n}$}})\\right)}-e^{P_j\\cL P_j}}_\\infty\\\\\n\t\t\t\t&\\qquad\\leq\\max\\limits_{s\\in[0,1]}\\norm{e^{sP_j\\cL P_j}}_\\infty\\norm{e^{(1-s)n\\left(C_{j}(\\text{\\tiny{$\\tfrac{1}{n}$}})-P_{j}(\\text{\\tiny{$\\tfrac{1}{n}$}})\\right)}}_\\infty\\norm{n\\left(C_{j}(\\tfrac{1}{n})-P_j(\\tfrac{1}{n})\\right)-P_j\\cL P_j}_\\infty\\\\\n\t\t\t\t&\\qquad\\leq\\frac{1}{n}\\max\\limits_{s\\in[0,1]}\\norm{e^{sP_j\\cL P_j}}_\\infty e^{w_j}b^2d_jr\\left(R_j^2\\left(d_jr+R_jr+2\\right)+\\frac{1}{2}d_jr+R_jd_jr+R_j\\right)\n\t\t\t\\end{align*}\n\t\t\tfor all $j\\in\\{1,..J\\}$ and where $w_j$ is defined in \\Cref{eq:proofthm2term2-approx}. With one more application of \\Cref{eq:profthm2term2-approx-constant0}, this shows\n\t\t\t\\begin{equation*}\n\t\t\t\t\\norm{\\sum_{j=1}^{J}\\lambda_j^ne^{n\\left(C_{j}(\\text{\\tiny{$\\tfrac{1}{n}$}})-P_{j}(\\text{\\tiny{$\\tfrac{1}{n}$}})\\right)}P_{j}(\\tfrac{1}{n})-\\sum_{j=1}^{J}\\lambda_j^ne^{P_j\\cL P_j}P_j}_\\infty\\leq\\frac{J}{n}e^{t\\tilde{d}_2}\\max\\limits_{s\\in[0,1]}\\norm{e^{stP_j\\cL P_j}}_\\infty\n\t\t\t\\end{equation*}\n\t\t\twhere we choose $\\tilde{d}_2\\geq0$ appropriately and redefine $\\cL$ by $t\\cL$ and $b$ by $bt$.\n\t\t\\end{proof}\n\t\n\t\t\\subsubsection*{End of the Proof of \\Cref{thm:spectral-gap-uniform}:}\n\t\tFinally, we combine the upper bounds found in the lemmas in order to finish the proof of \\Cref{thm:spectral-gap-uniform}.\n\t\t\\begin{proof}[Proof of \\Cref{thm:spectral-gap-uniform}]\n\t\t\t\\Cref{lem:proofthm2-term1}, \\ref{lem:proofthm2-term2}, and \\ref{lem:proofthm2-term3} show for all $t\\in[0,n\\epsilon]$ with $\\epsilon\\coloneqq \\min\\{\\epsilon_1,\\epsilon_2\\}$\n\t\t\t\\begin{flalign*}\n\t\t\t\t&&\\norm{\\left(Me^{\\frac{t}{n}\\cL}\\right)^n-\\sum_{j=1}^{J}\\lambda_j^ne^{tP_j\\cL P_j}P_j}_\\infty&\\leq c_2\\tilde{\\delta}^n+\\frac{tb}{n}+\\frac{tb}{n}\\frac{c_{p}c_2(2+tbc_{p}c_2)(\\tilde{\\delta}-\\tilde{\\delta}^{n})}{1-\\tilde{\\delta}}e^{2tbc_{p}c_2}&&\\text{(\\Caref{lem:proofthm2-term1})}\\\\\n\t\t\t\t&& &\\quad+\\frac{J}{n}e^{t\\tilde{d}_1}&&\\text{(\\Caref{lem:proofthm2-term2})}\\\\\n\t\t\t\t&& &\\quad+\\frac{J}{n}e^{t\\tilde{d}_2}\\max\\limits_{s\\in[0,1]}\\norm{e^{stP_j\\cL P_j}}_\\infty&&\\text{(\\Caref{lem:proofthm2-term3})}.\n\t\t\t\\end{flalign*}\n\t\t\tFor an appropriate constant $c_1\\geq0$, we finish the proof of \\Cref{thm:spectral-gap-uniform}.\n\t\t\\end{proof}\n\t\n\t\n\t\\section{Examples}\\label{sec:applications}\n\t\tIn this section, we present two classes of examples, which illustrate the range of applicability of our results. In the examples, we denote by $\\rho,\\sigma$ quantum states. \n\t\t\\begin{ex}[Finite dimensional quantum systems]\\label{ex:finite-dim}\n\t\t\tWe choose $\\cX=\\cB(\\cH)$ to be the algebra of linear operators over a finite dimensional Hilbert space $\\cH$ endowed with the trace norm $\\|x\\|_1=\\tr|x|$, $M:\\cB(\\cH)\\to \\cB(\\cH)$ a quantum channel, i.e.~a completely positive, trace preserving linear map, and $\\cL$ the generator of a semigroup of quantum channels over $\\cB(\\cH)$, also known as a quantum dynamical semigroup. In finite dimension, it is know that every quantum channel is a contraction \\cite[Cor.~3.40]{Watrous.2018}, the spectrum includes the eigenvalue $1$ \\cite[Thm.~3]{Wolf.2010}, and every linear operator in finite dimension has a discrete spectrum. Moreover, the nilpotent part of a quantum channel is zero \\cite[Lem.~A.1]{Hasenohrl.2020}. Therefore, there exist $\\delta\\in(0,1)$, $\\tilde{c}>0$, and a set of eigenvalues and projections $\\{\\lambda_j,P_j\\}_{j=1}^J$ such that for all $n\\in\\N$\n\t\t\t\\begin{equation*}\n\t\t\t\t\\norm{M^nx-\\sum_{j=1}^J\\lambda_jP_jx}_1\\leq\\tilde{c}\\delta^n\\norm{x}_1\n\t\t\t\\end{equation*}\n\t\t Note that the assumptions on the semigroup are satisfied due to the finiteness of the system and the contraction property of the $P_j$ must be assumed additionally.\n\t\t\\end{ex}\n\t\tIn the following example class, we calculate $\\delta$ directly.\n\t\t\\begin{ex}[{Power convergence via strong data processing inequalities}]\n\t\t\t{As in \\Cref{ex:finite-dim}, we define $\\cX=\\cB(\\cH)$ endowed with the trace norm $\\|x\\|_1$, $M:\\cB(\\cH)\\to \\cB(\\cH)$ a quantum channel, and $\\cL$ the generator of a quantum dynamical semigroup.}\n\t\t\tHere, we further assume the existence of a projection $P:\\cB(\\cH)\\to \\mathcal{N}$ onto a subalgebra $\\mathcal{N}\\subset \\cB(\\cH)$ with $MP=PM$ and such that the following \\textit{strong data processing inequality} holds for some $\\hat{\\delta}\\in(0,1)$: for all states $\\rho\\in\\cX$,\n\t\t\t\\begin{align}\\label{SDPI}\n\t\t\t\tD(M(\\rho)\\|M\\circ P(\\rho))\\le \\hat{\\delta}\\,D(\\rho\\|P(\\rho))\\,,\n\t\t\t\\end{align}\n\t\t\twhere we recall that the relative entropy between two quantum states, i.e.~positive, trace-one operators on $\\cH$, is defined as $D(\\rho\\|\\sigma):=\\tr[\\rho\\log\\rho-\\rho\\log\\sigma]$, whenever $\\operatorname{supp}(\\rho)\\subseteq\\operatorname{supp}(\\sigma)$. Equation \\eqref{SDPI} was recently shown to hold under a certain detailed balance condition for $M$ in \\cite{gao2021complete}: there exists a full-rank state $\\sigma$ such that for any two $x,y\\in\\cB(\\cH)$, $$\\tr[\\sigma\\,x^*M^*(y)]=\\tr[\\sigma\\,M^*(x^*)y].$$ Here $x^*$, resp. $M^*$, denotes the adjoint of $x$ w.r.t.~the inner product on $\\cH$, resp. the adjoint of $M$ w.r.t. the Hilbert-Schmidt inner product on $\\cB(\\cH)$. In finite dimensions, the quantity $\\sup_\\rho\\,D(\\rho\\|P(\\rho))<\\infty$ is called the Pimsner-Popa index of $P$ \\cite{pimsner1986entropy}. Using Pinsker's inequality, we see that the assumption of \\Cref{thm:spectral-gap-uniform} is satisfied: for all $x=x^*\\in\\cB(\\cH)$ with $\\|x\\|_1\\le 1$ and decomposition $x=x_+-x_-$ into positive and negative parts and corresponding states $\\rho_{\\pm}=x_{\\pm}\/\\tr[x_{\\pm}]$,\n\t\t\t\\begin{align*}\n\t\t\t\t\\|(M^n-M^n\\circ P)(x)\\|_1& \\le \\tr[x_+]\\,\\|(M^n-M^n\\circ P)(\\rho_+)\\|_1+ \\tr[x_-]\\,\\|(M^n-M^n\\circ P)(\\rho_-)\\|_1\\\\\n\t\t\t\t&= \\|x\\|_1\\,\\sqrt{2}\\,\\max_{\\rho\\in\\{\\rho_+,\\rho_-\\}}D(M^n(\\rho)\\|M^n\\circ P(\\rho))^{\\frac{1}{2}}\\\\\n\t\t\t\t&\\le \\sqrt{2}\\,\\sup_\\rho\\,D(\\rho\\|P(\\rho))^{\\frac{1}{2}}\\,\\hat{\\delta}^{\\frac{n}{2}}\\eqqcolon \\tilde{c}\\,\\delta^n\\,.\n\t\t\t\\end{align*}\n\t\t\t{Then, we can apply \\Cref{cor:explicit-bound-prop} which proves that there is an $\\epsilon>0$ such that for all $n\\in\\N$, $t\\in[0,n\\epsilon]$, and $\\tilde{\\delta}\\in(\\delta,1)$\n\t\t\t\\begin{equation*}\n\t\t\t\t\\norm{\\left(Me^{\\frac{t}{n}\\cL}\\right)^n-e^{tP\\cL P}P}_\\infty=\\cO\\left(\\frac{e^{t\\|\\cL\\|_\\infty}}{n}+\\frac{\\tilde{\\delta}}{n}e^{\\frac{8\\tilde{c}t\\norm{\\cL}_\\infty}{\\tilde{\\delta}-\\delta}}\\right)\n\t\t\t\\end{equation*}\n\t\t\tfor $n\\rightarrow\\infty$.}\n\t\t\\end{ex}\n\t\t\\begin{ex}[Infinite dimensional quantum systems and unbounded generators]\n\t\t\tHere, we pick $\\cH=L^2(\\mathbb{R})$, denote by $I$ the identity operator on $\\cH$, let $\\sigma$ be a quantum state on $\\cH$ and $M$ a generalized depolarizing channel of depolarizing parameter $p\\in (\\frac{1}{2},1)$ and fixed point $\\sigma$:\n\t\t\t\\begin{align}\n\t\t\t\tM(\\rho):=(1-p)\\rho +p\\tr(\\rho)\\,\\sigma\\,.\n\t\t\t\\end{align}\n\t\t\tIt is clear by convexity that $M$ satisfies the uniform strong power convergence with projection $P(\\rho)=\\tr(\\rho)\\,\\sigma$ and parameter $\\delta=2(1-p)<1$,\n\t\t\t\\begin{equation*}\n\t\t\t \\norm{M(\\rho)-P(\\rho)}_1=(1-p)\\norm{\\rho -\\tr(\\rho)\\,\\sigma}_1\\leq2(1-p)\\norm{\\rho}_1.\n\t\t\t\\end{equation*}\n\t\t\tLet $\\cL$ be a generator of a $C_0$-contraction semigroup such that $\\sigma\\in\\cD(\\cL)$. For example, let $\\cH$ be the Fock-space spanned by the Fock basis $\\{\\ket{0},\\ket{1},\\ket{2},...\\}$, $a$ and $a^\\dagger$ be the annihilation and creation operator defined by $a\\ket{0}=0$, $a\\ket{j}=\\sqrt{j}\\ket{j-1}$ for all $j\\in\\N_{\\geq 1}$, and $a^\\dagger\\ket{j}=\\sqrt{j+1}\\ket{j+1}$ for all $j\\in\\N_{\\geq0}$. Then, define $e^{t\\cL}(\\rho)\\coloneqq e^{-itH}\\rho e^{itH}$ where $H=a^\\dagger a+\\tfrac{1}{2}$ ($H=-\\Delta+x^2$) is the Hamiltonian of the harmonic oscillator as in \\cite{Becker.2021} and \n\t\t\t\\begin{equation*}\n\t\t\t \\sigma=\\frac{1}{3}(\\ket{0}\\bra{0}+\\ket{1}\\bra{1}+\\ket{2}\\bra{2})+\\frac{1}{10}(\\ket{0}\\bra{1}+\\ket{1}\\bra{0}).\n\t\t\t\\end{equation*}\n\t\t\tThen, we have that for all $t\\ge 0$: \n\t\t\t\\begin{align}\\label{upperbound1}\n\t\t\t\t\\|(\\1-P)e^{t\\cL}P\\|_{1\\to 1}=\\sup_{\\|x\\|_1\\le 1}|\\tr(x)|\\,\\|e^{t\\cL}(\\sigma)-\\sigma\\|_1\\le t\\,\\|\\cL(\\sigma)\\|_1\\,.\n\t\t\t\\end{align}\n\t\t\tMoreover, by duality and the unitality of the maps $e^{t\\cL^*}$ we have that\n\t\t\t\\begin{align}\\label{upperbound2}\n\t\t\t\t\\|Pe^{t\\cL}(\\1-P)\\|_{1\\to 1}=\\|(\\1-P)^*e^{t\\cL^*}P^*\\|_{\\infty\\to\\infty}=\\sup_{\\|y\\|\\le 1}|\\tr(\\sigma y)|\\|(\\1-P)(I)\\|=0\\,,\n\t\t\t\\end{align}\n\t\t\tTherefore, the assumptions of \\Cref{thm:spectral-gap} are satisfied and we find the convergence rate $\\mathcal{O}(n^{-1})$. Interestingly, this answers a conjecture of \\cite[Ex.~3, 5]{Becker.2021} for the Hamiltonian evolution generated by the one-dimensional harmonic oscillator. There, the authors had numerically guessed the optimal rate which we prove here. However their analytic bounds could only provide a decay of order $\\mathcal{O}(n^{-\\frac{1}{4}})$ (q.v.~remark after \\Caref{lem:chernoff}) and for a restriction of $H$ to a finite dimensional stable subspace, which effectively assumed the boundedness of the generator.\n\t\t\\end{ex}\n The depolarizing noise considered in the previous example is artificial. In an infinite dimensional bosonic system, a more natural model of noise is the photon loss channel, which we consider in the next example.\n\n\n\t\t\\begin{ex}[Bosonic beam-splitter]\n We define the bosonic one-mode system by the algebra generated by the creation and annihilation operators $a^*$ and $a$ which satisfy the canonical commutation relation (CCR):\n\t\t\t\\begin{equation*}\n\t\t\t\t[a,a^*]=\\1\\,.\n\t\t\t\\end{equation*}\n\t\t\tThe associated Fock basis $\\{\\ket{0},\\ket{1},\\ket{2},...\\}$ is orthonormal and defined by \n\t\t\t\\begin{equation*}\n\t\t\ta^*\\ket{j}=\\sqrt{j+1}\\ket{j+1}\\quad\\text{and}\\quad a\\ket{j}=\\sqrt{j}\\ket{j-1}\n\t\t\t\\end{equation*}\n\t\t\twhere the vacuum state $\\ket{0}$ satisfies $a\\ket{0}=0$. The Fock basis spans a Hilbert space called Fock space on which the operators from the CCR algebra are defined. A bosonic quantum state is a semidefinite operator in the CCR algebra with trace 1. \n\t\t\tA bosonic $2$-mode system is defined by the CCR-algebra generated by $\\{a,b,a^*,b^*\\}$, which satisfy, additionally to the canonical commutation relation, $[a,b]=0$. Next, we consider the \\textit{quantum beam-splitter} for $\\lambda\\in [0,1)$ \n\t\t\t\\begin{equation*}\n\t\t\t\tM_\\lambda(y)\\coloneqq\\tr_2[U_\\lambda y\\otimes\\sigma U_\\lambda^*] \\,,\n\t\t\t\\end{equation*}\n\t\t\twhere $\\tr_2$ denotes the partial trace over the second register, $U_\\lambda\\coloneqq e^{(a^* b-b^* a)\\operatorname{arcos}(\\sqrt{\\lambda})}$, an environment state $\\sigma$, and $y$ an element in the CCR algebra generated by $\\{a^*, a\\}$. Moreover, $P(y)\\coloneqq\\tr[y]\\sigma$ defines a projection which satisfies $PM_\\lambda=M_\\lambda P=P$ with the adjoint $P^*(x)=\\tr[\\sigma x]\\1$.\n \n In order to establish e.g.~the uniform power convergence of \\Cref{thm:spectral-gap} in the topology of the trace distance, we would need to consider a convergence in the form of $\\|M_\\lambda^n(\\rho)-\\sigma\\|_1\\to 0$ in the limit of large $n$ and uniformly in the initial state $\\rho$. Such property is notoriously hard to prove even in the classical setting \\cite{PW16}. Instead, we will consider a different metric on the set of quantum states which turns out to be more easy to work with.\n\t\t\t\n\t\t\tWe write $\\cB_{{N}}$ for the linear space of all ${N}$-bounded operators, where $ N= a^\\dagger a$ corresponds to the photon number operator. That is the vector space of linear operators $X$ on $L^2(\\mathbb{R})$ such that for any $|\\psi\\rangle\\in \\operatorname{dom}(N)$, $|\\psi\\rangle\\in\\operatorname{dom}(X)$ and there are some positive constants $a,b$ such that\n\t\t\t\\begin{align*}\n\t\t\t \\|X|\\psi\\rangle\\|\\le a\\|N|\\psi\\rangle\\|+b\\|\\psi\\|\\,.\n\t\t\t\\end{align*}\n\t\t\t We define the \\textit{Bosonic Lipschitz constant} of a $X\\in \\cB_{{N}}$ as \\cite{Cambyse.2021}\n \\begin{align}\n \\|\\nabla X\\|^2 := \\sup_{|\\psi\\rangle,|\\varphi\\rangle}\\,|\\langle \\psi|[a,X]|\\varphi\\rangle |^2+|\\langle \\psi|[a^*,X]|\\varphi\\rangle|^2\\,,\\nonumber\n \\end{align}\n where the suppremum is over all pure states $|\\psi\\rangle,|\\varphi\\rangle\\in\\operatorname{dom}({N})$ of norm $1$. By duality, we then define the \\textit{Bosonic Wasserstein norm} of a linear functional $f$ over $\\mathcal{B}_N$ with $f(\\1)=0$ as\n \\begin{align*}\n \\|f\\|_{W_1}:=\\sup_{\\|\\nabla X\\|\\le 1}\\,\\big|f(X)\\big|\\,.\n \\end{align*}\n where the supremum is over all ${N}$-bounded, self-adjoint operators $X$. We then choose our Banach space $\\cX$ as the closure of the set of such linear functionals such that $\\|f\\|_{W_1}<\\infty$. In particular, whenever $f\\equiv f_{\\rho-\\sigma}$ is defined in terms of the difference between two quantum states $\\rho,\\sigma$ as $f_{\\rho-\\sigma}(X)=\\tr((\\rho-\\sigma) X)$, we denote the Wasserstein distance associated to the norm $\\|.\\|_{W_1}$ as (see also \\cite{Cambyse.2021}):\n \\begin{align*}\n W_1(\\rho,\\sigma):=\\|f_{\\rho-\\sigma}\\|_{W_1}\\,.\n \\end{align*}\t\t\t\n\t\t\tThese definitions extend the classical Lipschitz constant $\\|\\nabla f\\|:=\\sup_{x\\in\\mathbb{R}^2}|\\nabla f(x)|$ of a real, continuously differentiable function $f$ of $2$ variables as well as the dual Wasserstein distance over probability measures on $\\mathbb{R}^2$. \n\t\t\t\n\t\t\tIn order to relate the Wasserstein distance to the statistically more meaningful trace distance, we seek for an upper bound on the trace distance in terms of $W_{1}$. By duality of both metrics, this amounts to finding an upper bound on the Lipschitz constant $\\|\\nabla X\\|$ of any bounded operator $X$ in terms of its operator norm $\\|X\\|_\\infty$. However, a bound of that sort does not exist (as classically, one can easily think of bounded observables which are not \\textit{smooth}). In the classical setting, the problem can be handled by first \\textit{smoothing} the function $f$, e.g. by convolving it with a Gaussian density $g$. In that case, one proves that there exists a finite constant $C>0$ such that $\\|\\nabla (f\\ast g)\\|\\le C\\|f\\|_\\infty$. In analogy with the classical setting, we can prove that for any two states $\\rho_1,\\rho_2$ and $\\lambda\\in [0,1)$ (see also \\cite[Proposition 6.4]{Cambyse.2021}),\n\t\t\t\\begin{align}\\label{tracetoWass}\n\t\t\t \\|M_\\lambda(\\rho_1-\\rho_2)\\|_1\\le \\,C\\, W_{1}(\\rho_1,\\rho_2)\\,,\n\t\t\t\\end{align}\n\t\t\twhere $C^2:=(\\|[a,\\sigma]\\|_1^2+\\|[a^*,\\sigma]\\|_1^2)\\lambda {(1-\\lambda)^{-1}}$.\n\t\t\t\n With a slight abuse of notations, we also write $M_\\lambda(f)$ for $f\\circ \\mathcal{B}^*_\\lambda$. It remains to prove the uniform power convergence. Proposition 6.2 from \\cite{Cambyse.2021} gives \n\t\t\t\\begin{align*}\n\t\t\t\t\\norm{M_\\lambda(f)}_{W_1}&=\\sup_{\\|\\nabla X\\|\\leq1}\\abs{f\\circ M_\\lambda^*(X)}\\\\\n\t\t\t\t&=\\sup_{\\|\\nabla X\\|\\leq1}\\abs{f\\left(\\frac{M_\\lambda^*(X)}{\\|\\nabla M_\\lambda^*(X)\\|}\\right)}\\|\\nabla M_\\lambda^*(X)\\|\\\\\n\t\t\t\t&=\\sup_{\\|\\nabla X\\|\\leq1}\\abs{f(X)}\\sqrt{\\lambda}\\\\\n\t\t\t\t&=\\sqrt{\\lambda}\\,\\|f\\|_{W_1}\n\t\t\t\t\\end{align*}\n\t\t\t\tThe uniform power convergence follows by $P(f)(X)\\equiv f\\circ P^*(X)=\\tr(\\sigma X)f(\\1)=0$. Moreover, the asymptotic Zeno condition (\\ref{eq:thm1-asympzeno}) is satisfied if $\\sigma\\in\\cD(\\cL)$ so that \\Cref{thm:spectral-gap} is applicable.\n\t\t\\end{ex}\n\t\tAs illustrated here, our asymptotic Zeno condition is easily verifiable and provides a rich class of examples. More examples for which our optimal convergence rate holds can be found in \\cite{Becker.2021}.\n\t\t\n\t\t\n\t\\section{Discussion and Open Questions}\\label{sec:discussion}\n\t\tIn this paper, we proved the optimal convergence rate of the quantum Zeno effect in two results: \\Cref{thm:spectral-gap} focuses on weakening the assumptions of the $C_0$-semigroup to the so-called asymptotic Zeno condition. Hence, \\Cref{thm:spectral-gap} allows strongly continuous Zeno dynamics which is novel for open systems. In \\Cref{thm:spectral-gap-uniform} instead, we weaken the assumption on $M$ to the uniform power convergence as in \\cite[Thm.~3]{Becker.2021}. Additionally, we presented an example which shows the optimality of the achieved convergence rate. This brings up the question whether our assumptions are optimal and how the assumption on the contraction correlates with the assumption on the $C_0$-semigroup. For example, is it possible to weaken the uniform power convergence in \\Cref{thm:spectral-gap} or \\Cref{prop:spectral-gap-uniform-norm-power-convergence} to finitely many eigenvalues on the unit circle without assuming stronger assumption on the semigroup? Following our proof strategy (q.v.~\\Caref{lem:proofthm1-term1}), this question is related to the conjecture of a generalized version of \\textit{Trotter's product formula for finitely many projections} under certain assumptions on the generator,\n\t\t\\begin{equation*}\n\t\t\t\\norm{\\left(\\sum_{j=1}^{J}\\lambda_jP_je^{\\frac{1}{n}\\cL}\\right)^nx-\\sum_{j=1}^{J}\\left(\\lambda_jP_je^{\\frac{1}{n}\\cL}\\right)^nx}\\overset{?}{=}\\cO\\left(\\frac{1}{n}(\\norm{x}+\\norm{\\cL x}+\\|\\cL^2 x\\|)\\right),\n\t\t\\end{equation*}\t\t\n\t\tAnother line of generalization would be to weaken the assumption on $M$ to the strong topology as in Theorem 2 in \\cite{Becker.2021}. There, the authors assume that $M^n$ converges to $P$ in the strong topology and that the semigroup is uniformly continuous. It would be interesting to know whether an extension to $C_0$-semigroups is possible. \n\t\tFinally, another important line of generalization would be to extend our results to time-dependent semigroups as in \\cite{Mobus.2019}.\n\n\n\t\\emph{Acknowledgments:} We would like to thank Michael Wolf and Markus Hasen\\\"ohrl for their support on this project. Moreover, we would like to thank Valentin A. Zagrebnov for his helping out with questions on the Chernoff Lemma and the anonymous reviewers for their constructive and detailed feedback. T.M. and C.R. acknowledge the support of the Munich Center for Quantum Sciences and Technology, and C.R. that of the Humboldt Foundation.\n\t\n\n\t\\setlength{\\bibitemsep}{0.5ex}\n\t\\printbibliography[heading=bibnumbered]\n\t\\vspace{2ex}\n\t\\addresseshere\n\t","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\\label{introduction}\n\nPhotometric redshifts (Connolly {\\it et al.~}~1995, Hogg {\\it et al.~}~1998, Benitez\n2000) are of paramount importance for current and planned multi-band\nimaging surveys. With photometric redshifts, surveys can\ninexpensively gather information about structure along the line of\nsight, without resorting to expensive spectroscopic followup.\nTherefore, it is important to understand systematic errors and\nlimitations in this method. For example, Ma {\\it et al.~}~2006 and Huterer\n{\\it et al.~}~2006 have examined the required photometric redshift accuracy\nfor surveys which plan to use weak lensing (cosmic shear) to constrain\ndark energy. For this application and also for baryon acoustic\noscillations (Zhan \\& Knox 2006), reducing photometric redshift errors\nis less important than knowing the error distribution accurately.\nThus, careful attention must be paid to systematic differences between\nthe photometric survey and the spectroscopic sample used to evaluate\nphotometric redshift performance. For most surveys, photometric S\/N\nis one of the systematic differences.\n\nThe most well-known test case for photometric redshifts is the blind\ntest in the Hubble Deep Field North (HDFN) conducted by Hogg {\\it et al.~}\\\n(1998). The best methods then yielded $\\sigma_{{\\Delta z}}\\sim0.1$, where\n${\\Delta z}\\equiv {{z_{\\rm spec}} - {z_{\\rm phot}} \\over 1 + {z_{\\rm spec}}}$, using Hubble Space Telescope\n(HST) photometry in UBVI bands and ground JHK (Dickinson ~1998). More\nrecently, with improved photometry and spectral redshift\nclassification, an accuracy of $\\sigma{\\Delta z}\\sim0.06$ is achieved over\nthe redshift range 0---6 (Fernandez-Soto {\\it et al.~}~1999, 2001; Benitez\n2000). Ground-based surveys suffer from less precise photometry but\nusually do not have to deal with such a large redshift range. Ilbert\n{\\it et al.~}~2007 cite an accuracy of $\\sigma_{{\\Delta z}} = 0.029$ after clipping\noutliers with ${\\Delta z}>0.15$ (3.8\\% of the sample). Ilbert {\\it et al.~}~2007\nalso find a decrease in precision at fainter magnitudes, but made no\neffort to separate the effects of S\/N from the other effects operating\non faint galaxies, such as a weaker magnitude prior and greater SED\nevolution. In this paper, we examine the impact of these effects\nseparately, focusing on photometric S\/N. \nThe quantitative results presented here are specific to the BVRz\nfilter set used in the Deep Lens Survey ({DLS}, Wittman {\\it et al.~}~2002).\nMore filters, covering a wider range in wavelengths, will do better\n(Abdalla {\\it et al.~}~2007). However, the trends with S\/N are broadly\napplicable.\n\n\n\\section{Method}\\label{method}\n\nWe use the {\\it BPZ}~ Bayesian photometric redshift code developed by\nBenitez (2000). We also tested the HyperZ code (Bolzonella {\\it et al.~}\\\n2000) with additional priors roughly equivalent to the default BPZ\npriors, and found similar performance. For clarity we present only\nresults from BPZ here. We did not test training-set methods, in which\na spectroscopic and photometric training set is used to perform a fit\nor to train a neural network, for two reasons. First, training set\nmethods are unlikely to be employed for surveys planning to push the\nphotometric sample deeper than the spectroscopic sample. Second, the\ntwo methods seem to be roughly equivalent in performance on the data\nsets in which they have been compared (e.g. Hogg {\\it et al.~}~1998), so the\ntrends presented here should be applicable to both methods.\n\nWe use the six spectral energy distribution ({\\it SED}) templates from\nBenitez (2000): E, Sbc, Scd, Irr, SB3, and SB2, modified as described\nbelow. For the simulations, the same templates are used to simulate\nthe photometry and to infer the photometric redshifts; there is no\nallowance for cosmic variance of the templates or ``template noise''.\nFor the data, it is important that the templates reflect real SEDs.\nTherefore, we use the photometry of objects with spectroscopic\nredshifts to optimize the templates (Csabai {\\it et al.~}~2000; Benitez\n{\\it et al.~}~2004; Ilbert {\\it et al.~}~2007). Section~\\ref{tempopt} describes the\nprocedure and shows the corrected templates. Clearly, even the\noptimized templates do not represent all types of SEDs in the\nuniverse. For both simulations and data, we start by demonstrating the\nperformance with as nearly perfect a data set as possible. After\nillustrating the best-case scenarios, we proceed to degrade the\nsimulations and data to successively lower S\/N, repeating the analysis\nfor each step.\n\nFor each galaxy, we identify the peak of its redshift probability\ndensity function (PDF) as its {\\it photometric redshift} or $z_{\\rm\nphot}$. This greatly simplifies the analysis and presentation of the\nresults, at the cost of some precision. Specifically, ``catastrophic\noutliers'' will appear, whose $z_{\\rm phot}$ differs greatly from\ntheir true redshift. In many cases, this may be an artifact of not\nconsidering the full PDF, a point argued forcefully in the case of the\nHDF by Fernandez-Soto\n{\\it et al.~} (2001, 2002). The full PDF may contain additional peaks, or\notherwise be broad enough to be consistent with the true redshift. In\nthis paper, we wish to focus on the trends with photometric S\/N rather\nthan the characterization of outliers. As will be seen in the tables\nand figures, the trends with S\/N are not substantively changed if\n``outliers'' are removed. Therefore we judge this simplification to\nbe acceptable. ``Outlier'' in this paper thus refers to difference\nbetween $z_{\\rm phot}$ and true redshift, without implying anything\nabout the full PDF.\n\nWe do consider characteristics of the PDF when using BPZ's ODDS\nparameter. BPZ assumes a natural error (template noise) of\n$0.067(1+z)$, and defines ODDS as the fraction of the area enclosed by\nthe PDF between $zphot\\pm n\\times0.067(1+z)$, where $n$ is a\nuser-settable parameter which we set to 1. ODDS values close to unity\nindicate that most of the area under the redshift probability density\nfunction (PDF) is within $Z_B\\pm0.067(1+z)$. In this paper, we\npresent results both for the entirety of a given sample, and after a\ncut of $ODDS>0.9$, which eliminates many of the ``outliers.'' We\nalso investigate the tradeoff between ODDS cut, number of usable\ngalaxies, and photometric redshift accuracy.\n\nThe error distributions are typically non-Gaussian, often highly so.\nThe rms or standard deviation is extremely sensitive to even a few\nnon-Gaussian events, so in the photometric redshift literature,\nresults are usually quoted as an rms after excluding a certain (small)\nfraction of galaxies as ``catastrophic outliers.'' The fraction\nvaries from paper to paper, making comparison difficult. The field of\nrobust statistics suggests several less sensitive metrics of\nvariation, such as the median or mean absolute deviation. However,\noutliers {\\it should} be included in the performance analysis with\nsome weight, because they will be included when using the entire\nphotometric sample for science. We therefore clip conservatively,\n$|{\\Delta z}|<0.5$, to avoid overly optimistic results. This threshold\nis at least five, and usually many more, times the clipped rms. We\nalso present, in many cases, differential and cumulative distributions\nas well. To make the connection with forecasts for, say, weak lensing\ntomography, we suggest these distributions be fit with double\nGaussians. Gaussians are analytically tractable, and a double\nGaussian can fit both the core and wings (but not truly catastrophic\noutliers).\n\n\n\\section{Simulations}\\label{simulations}\n\nWe simulate a mix of ellipticals, spirals, irregulars and starburst\ngalaxies (specifically, E, Sbc, Scd, Irr, SB3, and SB2 templates)\nfollowing the priors for galaxy type fraction as a function of\nmagnitude, $P(T|m_0)$, and for the redshift distribution for galaxies\nof a given spectral type and magnitude, $P(z|T,m_0)$, that are used in\n{\\it BPZ} 's Bayesian photometric redshift code. \nWe found that in Table 1 of Benitez (2000), two numbers were\ninadvertently switched, but the numbers were correct in the publicly\ndownloadable code. Benitez (private communication) has confirmed that\nthe table should read $k_t=0.450$ for E\/SO and $k_t=0.147$ for\nSbc\/Scd. Figure~\\ref{fig-priors} shows in solid red lines the priors\nused in this paper (same as {\\it BPZ} 's code); in dashed red lines are the\npriors quoted in BPZ's paper; and green lines represent Ilbert\n{\\it et al.~}~(2007) priors.\n\nIn order to have a realistic galaxy luminosity function, $N(mag)$, we\nstart our simulations from R-band magnitudes of 87260 objects detected\nin one of our $\\sim40^{\\prime}\\times40^{\\prime}$ Deep Lens Survey\nsub-fields (Wittman {\\it et al.~} 2002). The typical BVRz magnitude\ndistributions for the DLS are shown in Figure~\\ref{fig-nmagdls}. We\ntake this magnitude as the {\\it true} ($\\ne observed$) R-band\nmagnitude of a new object to be simulated. From the $P(T|m_0)$ prior\nwe select a {\\it SED}, and from $P(z|T,m_0)$ we choose a ${z_{\\rm input}}$ redshift for\nthe galaxy. The resulting ``true'' redshift distribution in the\nsimulations is shown in Figure~\\ref{fig-nzinput}. This distribution\nhas a larger tail to high redshift than usually found in the\nliterature (e.g. LeFevre et al 2005) and can be approximately described\nas $z^{2} exp\\big(-1(\\frac{z}{0.05})^{0.54}\\big)$. Magnitudes (with or\nwithout noise) in any other photometric bands can then be computed. We\nuse {\\it BPZ}~itself to compute synthetic colors, so there is no issue of\nminor differences in the k-corrections, priors, etc. We assume that\nthere are only six {\\it SED} `s of galaxies in the universe and make no\nattempt to introduce template noise in these simulations. We then\nperform three sets of simulations in the BVRz filter set of the\nDLS. In the first simulation (SIM1) we assume perfect, infinite\n{S\/N}~photometry. In the second set of simulations (SIM2) we\nsuccessively degrade the {S\/N}~ of the photometry but maintain constant\nthe {S\/N}~of all galaxies in all 4 bands (same magnitude error for all\ngalaxies in all 4 bands). In the third simulation (SIM3), we reproduce\nthe {S\/N}~distribution and completeness of the DLS.\n\n\n\\subsection{SIM1}\\label{sim1}\n\nThe first simulation (SIM1) has perfect photometry and represents the\nbest possible case. The ${z_{\\rm phot}}-{z_{\\rm spec}}$ scatter-plot for this simple\nsimulation is shown in Figure~\\ref{fig-sim1}, and the distribution of\n${\\Delta z} \\equiv {{z_{\\rm spec}} - {z_{\\rm phot}} \\over 1 + {z_{\\rm spec}}}$ is shown in\nFigure~\\ref{fig-sim1dz}. Note that Figure~\\ref{fig-sim1} contains\n87260 objects, distributed in redshift according to\nFigure~\\ref{fig-nzinput}, and that the ${z_{\\rm phot}}={z_{\\rm spec}}$ line is saturated\nwith objects. It is clear from Figure~\\ref{fig-sim1dz} that the\nmajority of objects have $|{\\Delta z}|\\sim0.0$. Table~\\ref{tab:sim12}\nindicates: (1) signal-to-noise of photometry (same in all bands); (2)\nfraction of galaxies with $|{\\Delta z}|<0.5$; (3) mean ${\\Delta z}$ for galaxies\nwith $|{\\Delta z}|<0.5$; (4) rms in ${\\Delta z}$ for galaxies with $|{\\Delta z}|<0.5$; (5)\nfraction of objects with $ODDS>0.9$; (6) fraction of objects with\n$ODDS>0.9$ and $|{\\Delta z}|<0.5$; (7) mean ${\\Delta z}$; and (8) rms in ${\\Delta z}$ for\nthese galaxies. \n\nThere are still catastrophic outliers, despite being the best possible\ncase in terms of noise, perfectly known templates, etc. This is\nbecause each galaxy is assigned a single ${z_{\\rm phot}}$ based on the peak of\nits PDF. Consider a degeneracy such that the same colors come from\n{\\it SED}~ A at $z_1$ or {\\it SED}~ B at $z_2$. In the absence of priors, this\nwould result in a PDF with two equal peaks. Now add priors encoding\nour astrophysical knowledge, such as that an apparently bright galaxies\nare likely to be at low redshift, or that ellipticals are rare at high\nredshift. This usually helps select the correct peak, but sometimes\nit will select the wrong peak because unlikely events do happen: some\nhigh-redshift galaxies are bright, or are ellipticals. As noted\nabove, this ignores the full PDF, which may be broad or multi-modal in\na way that is consistent with the true redshift. As our purpose is\nonly to establish SIM1 as a baseline for investigating the impact of\nphotometric S\/N, we do not pursue this here.\n\n\n\\subsection{SIM2}\\label{sim2}\n\nIn the second set of simulations (SIM2) we degrade the initially\nperfect photometry in SIM1 successively to {S\/N}~ of 250 ($R\\sim20.5^m$\nin the DLS, and the magnitude limit of the spectroscopic sample\npresented in Section \\ref{data}), 100, 60, 30, 10 and 5, and repeat\nthe analysis at each step. In these unrealistic simulations all\ngalaxies have the same photometric {S\/N}~ in all bands. The\nscatter-plots are shown in Figure~\\ref{fig-sim2}, and ${\\Delta z}$\ndistributions are shown in Figure~\\ref{fig-sim2dz}. We also present\nthe cumulative fraction of objects with ${\\Delta z}$ smaller than a given\nvalue, as a function of ${\\Delta z}$ (Figure~\\ref{fig-sim2frac}). This plot\nhas several advantages. First, multiple simulations can be\nover-plotted without obscuration. Second, the asymmetry in the\ndistribution of ${\\Delta z}$ is easily read off by looking at the fraction\nwith ${\\Delta z}<0$ (dashed vertical line). Third, the fraction of outliers\ncan also be directly read off the plot at any ${\\Delta z}$. The left panel of\nFigure~\\ref{fig-sim2frac} shows the cumulative fraction for all\nobjects, while the right panel shows $ODDS>0.9$ galaxies. The number\nof galaxies in the right panel is smaller than the number in the left\n(see Table~\\ref{tab:sim12}) but the accuracy of photo-zs is clearly\nbetter.\n\nBecause all realizations of SIM2 have the redshift distribution shown\nin Figure~\\ref{fig-nzinput}, even if all galaxies have colors measured\nat very high {S\/N}~, some objects will have degenerate colors and the\nsample will contain some fraction of catastrophic\noutliers. Spectroscopic samples typically have a much lower mean\nredshift than these simulations, so catastrophic outliers are likely to\nbe underrepresented in direct ${z_{\\rm phot}}-{z_{\\rm spec}}$ comparisons, if the full\nphotometric sample is very deep.\n\nTable~\\ref{tab:sim12} presents the statistics for the SIM2 objects\nshown in Figures~\\ref{fig-sim2}, ~\\ref{fig-sim2dz} and\n~\\ref{fig-sim2frac}. Clearly, the precision of photometric redshifts\nis a strong function of photometric {S\/N}. BPZ's ODDS parameter is very\neffective at removing outliers, and almost 100\\% of the objects with\n$ODDS>0.9$ have $|{\\Delta z}|<0.1$ regardless of {S\/N}~(right panel in\nFigure~\\ref{fig-sim2frac}). However, the fraction of objects with\n$ODDS>0.9$ decreases dramatically with decreasing {S\/N}.\n\nPerformance is, counter-intuitively, slightly worse for the infinite\nS\/N galaxies in SIM1 than for the high S\/N galaxies in SIM2. This is\nbecause the priors have too much power when there is no noise in color\nspace, and is not of concern in more realistic situations.\n\n\n\\subsection{SIM3}\\label{sim3}\n\nThe third simulation has the same {S\/N}~ distribution and completeness as\nthe DLS data. Again, the priors used assure that the galaxy type\nmixture and redshift distribution should be close to the real\nuniverse. The idea is to measure how well we can recover true ${z_{\\rm input}}$\nredshifts for a realistic photometric data set. This simulation is\nstill optimistic because no template noise is added---we derive colors\nfrom the same six templates used in the determination of photometric\nredshifts. The effect of template noise will be presented in the real\ndata analysis in Section \\ref{data}.\n\nAs a sanity check we compare the BVz magnitude distributions of our\nSIM3 simulation with the observed $N(mag)$ and find good\nagreement. The R magnitude distribution is by definition the same\nwithin the added photometric noise. We also compare the distribution\nof BPZ galaxy types in DLS fields with the one derived from the SIM3\nsimulation and find very good agreement. Figure~\\ref{fig-priors2}\nshows the galaxy type fraction as a function of magnitude for two\n$40^{\\prime}\\times40^{\\prime}$ DLS fields. The field with the higher\nfraction of ellipticals contains the richness class 2 galaxy cluster\nAbell 781 at $z=0.298$ (``$+$''), and the other is a more typical\n``blank'' field (``$\\times$''). The simulation input distribution is\nindicated by solid circles, which by definition agree with the red\nline, and the output BPZ types are indicated by open circles. SIM3 and\ndata show the same magnitude dependence.\n\nA third sanity check is a comparison between the redshift distribution\nderived in SIM3 and $N(z)$ for the entire DLS survey.\nFigure~\\ref{fig-dlsnz} shows both distributions and also the input\nredshift distribution used in the simulations (same as shown in\nFigure~\\ref{fig-nzinput}). The agreement is pretty good. The mean\ndensity of galaxies with photometric redshifts of any quality is\n$47\/arcmin^2$ and $11\\%$ of those objects have $ODDS>0.9$.\n\nThe photometric redshift performance on SIM3 is shown on\nFigures~\\ref{fig-sim3}, ~\\ref{fig-sim3dz} and~\\ref{fig-sim3frac}, just\nas in Figures~\\ref{fig-sim2}, ~\\ref{fig-sim2dz} and~\\ref{fig-sim2frac}\nfor SIM2. The summary statistics for SIM3 are presented in\nTable~\\ref{tab:sim3}. As in SIM2, the precision of photometric\nredshifts is a strong function of {S\/N}, and ODDS does a good job of\ncleaning up, at the cost of losing many low {S\/N}~galaxies.\n\nThere are two notable differences with SIM2. First, in SIM3, there is\na realistically strong correlation between high S\/N and bright\nmagnitudes. A bright magnitude implies a strong prior (most bright\ngalaxies are at low redshift), whereas a faint galaxy has a weak prior\n(it could be at any redshift). The high S\/N galaxies in SIM2 were\n(artificially) at all magnitudes, and therefore had generally looser\npriors. Therefore, the highest S\/N galaxies in SIM3 do better than\nthose in SIM2. We can see the effect of the tight priors directly by\ncomparing the $S\/N=250$ line of Table ~\\ref{tab:sim12}\n($\\sigma_{{\\Delta z}}=0.042$ after clipping 4\\% which had $|{\\Delta z}|>0.5$) with\nthat of Table~\\ref{tab:sim3} ($\\sigma_{{\\Delta z}}=0.031$ with no need to\nclip any outliers). This difference vanishes when low S\/N galaxies\nfrom SIM3 are included.\n\nIn fact, the $S\/N=5$ galaxies in SIM2 outperform the $S\/N>5$ galaxies\nin SIM3, despite the latter cut being only a lower bound. This is due\nto the second salient difference between SIM2 and SIM3: A given S\/N in\nSIM2 describes {\\it each} galaxy in {\\it each} band. In SIM3, the S\/N\nvaries with filter in a realistic way, and the cut applies to R band.\nMost galaxies will have lower S\/N in other bands. For $S\/N=30$ in R,\nthe median $S\/N$ in B, V, and z over the whole sample is 10, 18, and\n10 respectively.\n\n\nWhat S\/N is required for good photometric redshift performance?\nFirst, consider performance without any ODDS cut. At each step in\nTable~\\ref{tab:sim3} from $S\/N>100$ to $S\/N>10$, there is a 30--50\\%\nincrease in $\\sigma_{{\\Delta z}}$, so there is no natural breakpoint.\n$\\sigma_{{\\Delta z}}$ appears to stop this dramatic growth when stepping down\nfrom $S\/N>10$ to $S\/N>5$, but this is likely an artifact of clipping\nat $|{\\Delta z}|>0.5$, which is roughly three times the clipped rms at that\npoint. Even at $S\/N>10$, $\\sigma_{{\\Delta z}}$ may be artificially low due to\nclipping, as more than 10\\% of galaxies were clipped. Most survey\nusers would find the precision offered by the $S\/N>30$ cut acceptable,\nbut the $S\/N>10$ cut unacceptable. If we set $\\sigma_{{\\Delta z}}=0.1$ as\nthe limit of acceptability, we find an S\/N cut at 17 is required.\n\nNow consider using the ODDS cut at 0.9. $\\sigma_{{\\Delta z}}$ is always 0.04\nor less, regardless of S\/N. We suspect that for a given\n$\\sigma_{{\\Delta z}}$, the ODDS cut will provide more galaxies than the S\/N\ncut, because ODDS responds to the properties of the color space as\nwell as to S\/N. For example, high-precision S\/N is not required if\nthe galaxy is in a distinctive region of color space. In addition,\nODDS can take proper account of different S\/N in different bands,\nwhich a simple S\/N cut in R does not. We investigate this by finding\nthe ODDS cut which yields the same $\\sigma_{{\\Delta z}}$ as the $S\/N>30$ cut\n(0.076). We find that $ODDS>0.57$ is required, which yields 30\\% of\nall detected galaxies, vs. the 13\\% yielded by the S\/N cut.\n\nWe repeat this procedure for $\\sigma_{{\\Delta z}}=0.1$. The required ODDS\ncut is $>0.40$, yielding 45\\% of all detected galaxies, while the\nrequired S\/N cut at 17 yields only 26\\% of detected galaxies.\n\nThese fractions can all be read off\nFigure~\\ref{fig-sim3snodds} which summarizes the results from\nSIM3. The three left panels in Figure~\\ref{fig-sim3snodds} show: (1)\nthe cumulative fraction of objects with {S\/N}~greater than a given\nvalue; (2) mean ${\\Delta z}$; and (3) $\\sigma_{{\\Delta z}}$ for these objects. The\nthree right panels are the same but for a cut in $ODDS$.\n\nIn short, we recommend an ODDS cut. We recognize that an ODDS cut is\nnot easy to incorporate into survey forecasts of the number of usable\ngalaxies. Detailed simulations for a given filter set and depth as a\nfunction of wavelength must be performed. However, we hope that the\nabove numbers can serve as a rough guide for translation between\nphotometric redshift precision, S\/N threshold, and number of usable\ngalaxies.\n\n\n\\section{Data}\\label{data}\n\nWe take photometric data from the {DLS}~BVRz full-depth images in\nfields with spectroscopic redshifts from the {\\it Smithsonian\nHectospec Lensing Survey} (SHeLS, Geller {\\it et al.~}\\ 2005), and from the\n{\\it Caltech Faint Galaxy Redshift Survey} (CFGRS, Cohen {\\it et al.~}~1999)\nsurveys. Here, by definition, template noise is present. In Sections\n\\ref{shels} and \\ref{cfgrs} we present the spectroscopic data and the\nphotometric redshift accuracy for these two samples, but before that\nwe present our methodology for color measurement (Section\n\\ref{colors}), and template optimization (Section \\ref{tempopt}).\n\n\n\\subsection{Measuring Colors}\\label{colors}\n\nWe performed simulations to determine the best photometry method in\nthe face of different point-spread function (PSF) sizes in the\ndifferent filters. We added galaxies with De Vaucouleurs (elliptical)\nand exponential disk (spirals) light profiles to the {DLS}~BVRz data\nusing standard IRAF-Artdata routines, ran SExtractor (Bertin \\&\nArnouts 1996) and measured colors with many different types of\nmagnitudes. Figure~\\ref{fig-magRmagerr} shows the results for\ngalaxies added to the R images. The B, V and z results are\nqualitatively the same, but because of differences in {S\/N}~and PSF\nthere is a shift in the magnitude axis, and slightly different\nscatter. The left panels show the results using $MAG_{iso}$ and right\npanels show $MAG_{auto}$. The top panels show the difference between\nmeasured $MAG$ and input $MAG_{input}$. De Vaucoleurs galaxies are\nmeasured to be $\\sim0.15^m$ fainter than their true magnitudes both by\n$MAG_{iso}$ and $MAG_{auto}$. The bottom panels show the distribution\nof $(MAG-MAG_{input})\/MAGerr$ as a function of magnitude. As noted by\nBenitez {\\it et al.~}~(2004), $MAG_{auto}$ gives better results for\nmagnitudes, but for photometric redshifts we are interested in good\ncolors as deep as possible.\n\nFigures~\\ref{fig-colorerr} and \\ref{fig-cc} \nshow the distribution of {\\it color} errors,\nwhich, for photometric redshifts, are more important than magnitude\nerrors. Again, $MAG_{iso}$ is on the left and $MAG_{auto}$ on the\nright. The systematic magnitude errors tend to cancel when\nconsidering colors, and $MAG_{iso}$ is now slightly better. It is\nimportant to note that the errors in magnitude errors are not driven\nby faint galaxies, and that in fact the discrepancies between real and\nestimated colors errors are significantly worse for bright\nobjects. \n\nIn summary, $MAG_{iso}$ gives slightly more precise colors at a given\nmagnitude. This translates to more galaxies being detected above a\ngiven S\/N threshold, providing another benefit. However, for either\n$MAG_{auto}$ or $MAG_{iso}$, the error estimates provided by\nSExtractor are optimistic, especially at the bright end. The solid\nlines in Figure~\\ref{fig-frac} show the cumulative fraction of objects\nas a function of magnitude and color error, normalized by the nominal\nerror from SExtractor. Much less than $68(95)\\%$ of the galaxies have\nactual errors within the nominal 1(2)$\\sigma$ magnitude error. Actual\ncolor errors are closer to nominal, but still optimistic. (Caveat:\nunlike most real galaxies, the simulated galaxies had zero color.)\nFrom this analysis we determine an ad hoc correction to the magnitude\nerrors estimated by SExtractor: we first multiply $MAGerr_{iso}$ by\n$1.5$, and then add in quadrature an error of $0.02^m$. The dashed\nlines in both panels of Figure~\\ref{fig-frac} show the results of this\ncorrection. This single correction puts the 68th and 95th percentiles\nof all the color distributions in the correct place, with the\nexception of the 68th percentile of $R-z$ color. This adjustment to\nthe magnitude errors should in principle depend on galaxy color, but\nwe found that variations about this correction made little difference\nin the results.\n\nWe performed all the real-data tests in this paper with both\n$MAG_{iso}$ and $MAG_{auto}$. The differences in the results were\nvery minor, except that more galaxies were detected at a given S\/N\nwith $MAG_{iso}$, and about 20\\% more survived the ODDS cut with\n$MAG_{iso}$. We therefore adopt $MAG_{iso}$ for the remainder of this\npaper.\n\nAnother factor to consider is the quality of the survey's photometric\ncalibration, which was determined by observations of standard stars in\nLandolt's (1992) fields during photometric nights. The R and V DLS\nbands are very similar to Landolt's filter transmissions and yield\naccurate calibration. The DLS B-band however differs significantly\nfrom Landolt's and requires a color term correction which decreases\nthe accuracy of calibration in this band. Also, the DLS z-band\nphotometry derived from Sloan Digital Sky Survey standards (Smith\n{\\it et al.~}~2002) is also not as good as R and V. For this reason we add an\nextra $0.01^m$ to the magnitude error measurements in B and z\nbands. \n\n\n\\subsection{Template Optimization}\\label{tempopt}\n\nWe use spectroscopic redshifts and the DLS photometry to empirically\ncorrect the {\\it BPZ}~set of templates and to test our filter+instrument\nresponse knowledge with the methodology described in Ilbert {\\it et al.~}\n2007. We find optimized templates for El, Sbc, Scd, Im, and SB3 {\\it SED}\ns. The SB2 template was left unchanged because there were not enough\ngalaxies of this type to fit a correction. The biggest modifications\nwere found for the El {\\it SED}, which shows a less strong 4000\\AA~ break\nin the optimized template; and for the Sbc {\\it SED}, which has a stronger\n4000\\AA~ break than in the original BPZ template (See\nFigure~\\ref{fig-seds}). Because most of our galaxies are at low\nredshift, we cannot constrain the longest and shortest SED wavelengths\nand therefore we force them to agree with the initial templates.\n\n\n\\subsection{Comparison with Spectroscopic Data: SHeLS Survey}\\label{shels}\n\nThe SHeLS survey has a limiting magnitude of $R=20.3$, so that the DLS\nphotometry, which is complete to about five magnitudes fainter, is\nvery high {S\/N}. Being a bright magnitude-limited survey, SHeLS\ncontains overwhelmingly low-redshift ($z<0.6$) galaxies. However, our\nsubsample of 1,000 was chosen to provide a nearly uniform redshift\ndistribution so that characterization accuracy would be roughly\nredshift-independent. At a given redshift, selection was random.\n\nWe further cut the sample, requiring ${S\/N}>100$ in the R band, and\nexcluding objects in exclusion zones around bright stars, or with\nsaturated pixels in any band, or with SExtractor $flags\\ge4$\n(compromised photometry). The final sample contains 860 galaxies.\nThe top left panels of Figures~\\ref{fig-shelsdatasn} and\n~\\ref{fig-shelsdatasndz} show the ${z_{\\rm phot}}-{z_{\\rm spec}}$ scatter-plot, and ${\\Delta z}$\ndistribution for the maximum {S\/N}~ photometry. The distribution of\ngalaxy types assigned by BPZ to this spectroscopic sample is in\nagreement to the type distribution of all galaxies at $R=20\\pm0.5^{m}$\nin the entire DLS survey, suggesting that the spectroscopic sample is\nrepresentative of galaxies at this magnitude.\n\nThe SHeLS sample is expected to show evidence of template noise and\nhave somewhat higher $\\sigma_{{\\Delta z}}$ than the bright end of SIM3, and\nthis is in fact observed. Objects with ${S\/N}>100$ in SIM3 have\n$\\sigma_{{\\Delta z}}=0.037$, and $89.4\\%$ of the galaxies have $ODDS>0.9$\nwith $\\sigma_{{\\Delta z}}=0.026$. For the SHeLS survey, $\\sigma_{{\\Delta z}}=0.050$,\nand $85.6\\%$ have $ODDS>0.9$ with $\\sigma_{{\\Delta z}}=0.044$. The difference\nsuggests a template noise of $\\sigma_{{\\Delta z}}\\sim0.035(1+z)$ which is\nsmaller than the $0.065(1+z)$ estimated by Fernandez-Soto {\\it et al.~}~(1999)\nfor galaxies in the Hubble Deep Field, but expected given the much\nlower redshift of galaxies in the SHeLS survey.\n\nWe now degrade the photometry successively to ${S\/N}=100,60,30,10,5$ in\nall bands. If a galaxy has, for example ${S\/N}=50$ in the B band, its\nmagnitude and magnitude error are left unchanged in this band for the\nsimulations with ${S\/N}=100$ and ${S\/N}=60$, but noise is added to the\nother ones. The ${z_{\\rm phot}}-{z_{\\rm spec}}$ scatter-plots are shown in\nFigure~\\ref{fig-shelsdatasn}. ${\\Delta z}$ distributions are shown in\nFigure~\\ref{fig-shelsdatasndz}, and cumulative fraction as a function\nof ${\\Delta z}$ is shown in Figure~\\ref{fig-shelsfrac}. Statistics in\ndifferent {S\/N}~regimes are presented in Table~\\ref{tab:shelsdata}.\nThe trends with S\/N which were observed in the simulations are\nreproduced here. \n\nBecause the magnitude prior remains tight despite the {S\/N}~degradation,\nwe observe lower $\\sigma_{{\\Delta z}}$ at the low {S\/N}~end of the SHeLS\nsimulations than is observed for SIM2 at the same {S\/N}. At ${S\/N}=10$,\n$\\sigma_{{\\Delta z}}=0.080$, and $8.3\\%$ of galaxies in the SHeLS survey have\n$ODDS>0.9$, while $\\sigma_{{\\Delta z}}=0.121$, and $6.4\\%$ of the have\n$ODDS>0.9$ for the SIM2 galaxies.\n\nThe effectiveness of the ODDS cut is again evident. The fraction of\ngalaxies passing this cut at low S\/N is less than in SIM3 because the\ndata here are uniformly at low S\/N, whereas for SIM3 the given S\/N is\na lower limit. The fraction with $ODDS>0.9$ at low S\/N is more\ndirectly comparable with, and more consistent with, the fractions in\nSIM2, which were also at constant S\/N.\n\n\n\\subsection{Comparison with Spectroscopic Data: CFGRS Survey}\\label{cfgrs}\n\nThe CFGRS (Cohen {\\it et al.~}~1999) survey is about $2^m$ deeper than SHeLS\nand therefore the DLS photometry is not as high {S\/N}. We select\ngalaxies with quality=1 (multiple spectral features, Cohen {\\it et al.~}~1999)\nspectroscopic redshifts and divide the data in 2 equally sized\nsubsamples of 111 galaxies each: one with galaxies of photometric\n${S\/N}(R)>106$, and another with ${S\/N}(R)<106$. Note that the\nsignal-to-noise in the low\n{S\/N}~ sample is still fairly high, with 28 being the lowest value, and\na median of 69, but the difference in the quality of photometric\nredshifts is clear. Figure~\\ref{fig-cfgrs} shows the ${z_{\\rm phot}}-{z_{\\rm spec}}$\nscatter-plot for the two sub-samples. For the high {S\/N}~sample,\n${\\Delta z}=0.027\\pm0.084$, and ${\\Delta z}=0.021\\pm0.060$ if we exclude 1\ncatastrophic outlier with $|{\\Delta z}|>0.5$. For the lower\n{S\/N}~sample, ${\\Delta z}=0.033\\pm0.166$, and ${\\Delta z}=0.041\\pm0.095$ if we exclude\n2 objects with $|{\\Delta z}|>0.5$. However this includes the effect of\ndifferent redshift ranges. To isolate the\n{S\/N}~effect, we compute results using only galaxies between\n$0.40.5$\nare found in this redshift range.\n\n\n\n\\section{Selection in Galaxy Type and Redshift Range}\n\nFigure~\\ref{fig-priors2} suggests that faint Irr\/SB2\/SB3 galaxies are\noften misclassified as Sbc\/Scd. In this section we explore dependence\non type in more detail. Figures~\\ref{fig-sim1typ},\n~\\ref{fig-sim3typ}, and ~\\ref{fig-datatyp} show the ${z_{\\rm phot}}-{z_{\\rm spec}}$\nscatter-plot as a function of inferred {\\it BPZ}~galaxy type ($T_B$) for\nSIM1, SIM3, and the SHeLS galaxies respectively. Ellipticals form the\ntightest relation, while the redshift of irregular galaxies show a\nscatter more than twice as large. Figure~\\ref{fig-sim3typ} shows that\nsome of the scatter in ellipticals must be due to misclassifications,\nbecause there are no E-type galaxies at $z\\sim3-4$ in the simulations.\n\nWe look at type misclassification in SIM3 directly in\nFigures~\\ref{fig-types} and ~\\ref{fig-typesall}. The left column of\npanels shows the $T_B$ distribution for each of the true input types,\nwith the true type distribution overlaid like a diagonal matrix in red\nto guide the eye. The right column of panels shows the true type\ndistribution for each of the inferred types, with the inferred type\ndistribution overlaid in red to guide the eye. The overall\ndistribution of inferred (true) types is shown by the unshaded\nhistogram which is repeated in each panel in the left (right) column.\nFigures~\\ref{fig-types} shows galaxies with ${S\/N}\\ge30$ or $R\\le23$.\nFor example, the fourth panel down in the left column shows that\ngalaxies classified as $T_B=4$ (Irr), have in fact almost the same\nprobability of being of types 4, 5 or 6 (irregular or starburst).\nLikewise, starburst galaxies tend to be misclassified at irregulars\neven at high {S\/N}.\n\nThe types in decreasing order of reliability are E, Sbc, Scd, Irr,\nSB3, and SB2. Type reliability translates to redshift reliability,\nbecause type misclassification usually implies a large, if not\ncatastrophic, redshift error. These figures also demonstrate that\nalthough the ODDS cut appears to lose many high high-redshift galaxies\nand shrink the usable redshift range, in fact most of the\n``high-redshift'' galaxies lost were type misclassifications, and\ntherefore unreliable redshifts. Although the loss of these\n``high-redshift'' galaxies is painful if one wants as large a redshift\nrange as possible, it is necessary if one wants the sample to be\nreliable.\n\nIn Figure~\\ref{fig-typesall} we extend the analysis to lower\n{S\/N}~galaxies, and include all ``detected'' galaxies. The\nrate of misclassification is much higher. The insertion of these\nobjects in the sample creates new types of misclassification. For\nexample, a fraction of type 1 (E) galaxies is assigned $T_B=2$ and\nvice-versa. Also, a significant fraction of types 4, 5, and 6\n(irregular and starburst) are classified as types 2 or 3 (spirals).\n\n\n\\section{Summary and Discussion}\n\nWe have examined the dependence of photometric redshift performance on\nphotometric S\/N, using both simulations and data. For concreteness,\nwe have used the DLS filter set, but the general trends should apply\nto any filter set. As a reminder, SIM2 simulated galaxies at a range\nof magnitudes drawn from the DLS photometry, but at a series of\nconstant S\/N levels, while SIM3 strongly couples magnitude and S\/N as\nthey are in the DLS photometry. Thus, {\\it bright} is distinct from\n{\\it high S\/N} in SIM2 and in the noise-augmented SHeLS data because\n{\\it bright} implies a more effective magnitude prior. An additional\ndistinction between SIM3 and the other cases is that in SIM3 a given\nS\/N cut is performed in R, and for most galaxies that implies a lower\nS\/N in the other bands. For SIM2 and noise-augmented SHeLS data, a\ngiven S\/N describes each galaxy in each filter.\n\nWe therefore expect the smallest $\\sigma_{{\\Delta z}}$ for very high S\/N in\nSIM3, where the high S\/N galaxies automatically have a tight magnitude\nprior. This is what is observed, $\\sigma_z=0.031$ (0.037) for\n$S\/N>250$ (100) in SIM3. Degeneracies in color space determine this\nperformance limit, which is therefore highly filter-set dependent.\nHowever, it sets a baseline for what follows. At $S\/N=100$ in the\nSHeLS data, $\\sigma_{{\\Delta z}}$ is about 35\\% larger than this baseline,\nsuggesting a cosmic variance or template noise component of\n$\\sigma_{{\\Delta z}}=0.035(1+z)$. For SIM2, $\\sigma_{{\\Delta z}}$ is also about\n32\\% larger than this baseline, presumably due to the looser magnitude\npriors on average. The deeper the survey, the less effective the\nmagnitude prior, but performance is still quite good at this high S\/N.\n\nFrom this baseline, lowering the S\/N smoothly increases $\\sigma_{{\\Delta z}}$\nin SIM3, by 30--50\\% at each S\/N step in Table~\\ref{tab:sim3} until\n$\\sigma_{{\\Delta z}}$ is no longer trustworthy due to the clipping at\n$|{\\Delta z}|>0.5$. SIM2 degrades a bit more slowly due to its higher\nbaseline $\\sigma_{{\\Delta z}}$. The noise-augmented SHeLS data degrades even\nmore slowly, because magnitude priors always remain tight. Although\n$\\sigma_{{\\Delta z}}$ looks reasonably good even at $S\/N=5$ for the degraded\nSHeLS data, we expect SIM3 to be more representative of true\nperformance for this reason.\n\n\nSIM3 indicates that without an ODDS cut, $S\/N=17$ in R is likely to be\nthe lowest acceptable S\/N for reasonable photometric redshift\nperformance ($\\sigma_{{\\Delta z}}=0.1$) in a survey with the DLS\nspecifications (filter set and depth). A shallower survey may be able\nto go to lower S\/N because the magnitude prior remains helpful to\nlower S\/N in such a survey. In fact, the bright spectroscopic sample\nhas $\\sigma_{{\\Delta z}}<0.1$ even at $S\/N=5$, although we caution that this\nmeans $S\/N=5$ in {\\it each} filter. If we impose an ODDS cut rather\nthan an S\/N cut, $ODDS>0.40$ cut yields twice as many galaxies for the\nsame $\\sigma_{{\\Delta z}}$ as the $S\/N>17$ cut in R. Alternatively, survey\nusers could use ODDS to decrease $\\sigma_{{\\Delta z}}$ while sacrificing\ngalaxy counts; an $ODDS>0.9$ cut yields $\\sigma_{{\\Delta z}}=0.04$ averaged\nover all S\/N.\n\nWe caution that there are some unmodeled effects which, if included,\nwould result in a larger $\\sigma_{{\\Delta z}}$. First, template noise is not\nincluded in the simulations. $\\sigma_{{\\Delta z}}$ is larger in the SHeLS\ndata than in SIM3 for $S\/N>60$, which we attribute to template noise.\nTemplate noise becomes less important at lower photometric S\/N, but\nthe template noise in the SHeLS data may be artificially low. The\ntemplates were originally derived from bright galaxies like those in\nSHeLS, and further optimized on the SHeLS sample itself. A\nphotometric sample which pushes to higher redshift may thus incur more\ntemplate noise, and in fact Fernandez-Soto {\\it et al.~}~(1999) estimates\n$\\sigma_{{\\Delta z}}=0.065(1+z)$ for galaxies in the Hubble Deep Field.\n\nSecond, because galaxy counts are rising beyond the limiting magnitude\nfor detection, an additional source of photometry noise must be taken\ninto account. A source detected at S\/N of a few is much more likely\nto be an ``up-scattered'' fainter galaxy than a ``down-scattered''\nbrighter galaxy. As pointed out by Hogg \\& Turner (1998, hereafter\nHT98), this is distinct from Malmquist bias, which is the\nover-representation of high-{\\it luminosity} galaxies in a flux-limited\nsample. Although the resulting bias can be computed and corrected for\nif the galaxy count slope is known, the additional photometric\nuncertainty is unavoidable. In fact, HT98 conclude that ``sources\nidentified at signal-to-noise ratios of four or less are practically\nuseless.'' This source of noise was not reproduced in our\nsimulations, so extrapolation to $S\/N<5$ would be extremely dangerous.\nOur results for $S\/N=5$ are still valid if five is interpreted as the\neffective S\/N in the presence of this additional source of noise. For\nthe no-evolution, Euclidean slope of $q=1.5$, the HT98 formulae\nindicate that this requires a detection at $S\/N=5.64$. For $S\/N=10$\nand higher, the corrections are very small.\n\nIn addition to these dependences on S\/N, several other lessons can be\ndrawn:\n\\begin{itemize}\n\n\\item When forecasting photometric redshift performance for a survey,\nit is important to include realistic photometry errors. \n\n\\item Estimating photometric redshift performance with spectroscopic\nsamples can lead to optimistic results if the spectroscopic sample is\nnot representative of the photometric sample. If the spectroscopic\nsample is brighter, matching the S\/N is easily accomplished by adding\nphotometry noise, but accounting for the larger redshift range of the\nphotometric sample requires detailed modeling which must account for\ncosmic variance.\n\n\\item The BPZ $ODDS$ parameter is very effective at identifying\nphotometric redshifts which are likely to be poor. An $ODDS$ cut is\nmore efficient than an S\/N cut, because $ODDS$ takes account of the\nlooser photometry requirements in distinctive regions of color space.\nStill, our simulations and artificially noisy data show that of the\ngalaxies with $ODDS<0.9$, the ones with poor photometric redshifts may\nbe in the minority. The tradeoff between $ODDS$ cut and usable\nnumbers of galaxies must be assessed in light of the specific science\ngoal. For example, if the science analysis weights each galaxy by its\nphotometric S\/N, a strict $ODDS$ cut may cut most of the galaxies but\nnot most of the total weight. For weak lensing, shape noise limits\nthe maximum weight of a galaxy, so a strict $ODDS$ cut may cut most of\nthe weight. Finally, biases must be considered, as ellipticals are\noverrepresented in the set of galaxies with high $ODDS$. This may not\naffect weak lensing but will be important for studies of galaxy\nevolution and baryon acoustic oscillations.\n\n\\end{itemize}\n\nWe also explored cutting in type (as identified by BPZ) and redshift\nrange. As expected, ellipticals do better than any other type, but we\nfound that the $ODDS$ cut was still useful for ellipticals. As long\nas the $ODDS$ cut was being used, other types could safely be used as\nwell. Therefore, we recommend cutting on ODDS rather than type. \n\n\n\\acknowledgments\n\nWe thank NOAO for supporting survey programs and the CFGRS project for\nmaking data publicly available. DLS observations were obtained at\nCerro Tololo Inter-American Observatory (CTIO) and Kitt Peak National\nObservatory (KPNO). CTIO and KPNO are part of the National Optical\nAstronomy Observatory (NOAO), which is operated by the Association of\nUniversities for Research in Astronomy, Inc., under cooperative\nagreement with the National Science Foundation. We also would like to\nthank Margaret Geller and Michael Kurtz for providing us with 1,000\nSHeLS redshifts, which were observed with Hectospec at the MMT\nTelescope.\n\nWe thank Ian Dell'Antonio and Tony Tyson for comments that led to\nimprovements to the paper.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\nDuring the past few years, the theory of frames have been growing\nrapidly and new topics about them are discovered almost every year.\nFor example, generalized frames (or g-frames), subspaces of frames\n(or fusion frames), continuous frames (or c-frames), $k$-frames,\ncontrolled frames and the combination of each two of them, lead\n to c-fusion frames, g-c-frames, c-g-frames, c$k$-frames,\nc$k$-fusion frames and etc. The purpose of this paper is to\nintroduce and review some of the generalized fusion frames (or g-fusion\nframes) and their operators. Then, we will get some useful\npropositions about these frames and finally, we will study g-fusion\nframe sequences.\n\nThroughout this paper, $H$ and $K$ are separable Hilbert spaces and $\\mathcal{B}(H,K)$ is the collection of all the bounded linear operators of $H$ into $K$. If $K=H$, then $\\mathcal{B}(H,H)$ will be denoted by $\\mathcal{B}(H)$. Also, $\\pi_{V}$ is the orthogonal projection from $H$ onto a closed subspace $V\\subset H$ and $\\lbrace H_j\\rbrace_{j\\in\\Bbb J}$ is a sequence of Hilbert spaces where $\\Bbb J$ is a subset of $\\Bbb Z$. It is easy to check that if $u\\in\\mathcal{B}(H)$ is an invertible operator, then (\\cite{ga})\n$$\\pi_{uV}u\\pi_{V}=u\\pi_{V}.$$\n\\begin{definition}\\textbf{(frame)}.\nLet $\\{f_j\\}_{j\\in\\Bbb J}$ be a sequence of members of $H$. We say that $\\{f_j\\}_{j\\in\\Bbb J}$ is a frame for $H$ if there exists $00$ for any $j\\in\\Bbb J$). We say that $(W_j, v_j)$ is a fusion frame for $H$ if there exists $00$ and $\\Lambda_j\\in\\mathcal{B}(H,H_j)$ for each $j\\in\\Bbb J$. We say $\\Lambda:=(W_j, \\Lambda_j, v_j)$ is a \\textit{generalized fusion frame} (or \\textit{g-fusion frame} ) for $H$ if there exists $00$. Now, we can write for each $f\\in H$\n$$\\sum_{j\\in\\Bbb J}v_j^2\\Vert \\Lambda_j \\pi_{W_j}f\\Vert^2=\\langle S_{\\Lambda}f, f\\rangle\\geq C\\Vert f\\Vert^2$$\nand\n$$\\sum_{j\\in\\Bbb J}v_j^2\\Vert \\Lambda_j \\pi_{W_j}f\\Vert^2=\\langle S_{\\Lambda}f, f\\rangle\\leq \\Vert S_{\\Lambda}\\Vert \\Vert f\\Vert^2.$$\nIt follows that $\\Lambda$ is a g-fusion frame for $H$.\n\\end{proof}\n\\begin{theorem}\nLet $\\Lambda:=(W_j, \\Lambda_j, v_j)$ and $\\Theta:=(W_j, \\Theta_j, v_j)$ be two g-fusion Bessel sequence for $H$ with bounds $B_1$ and $B_2$, respectively. Suppose that $T_{\\Lambda}$ and $T_{\\Theta}$ be their analysis operators such that $T_{\\Theta}T^*_{\\Lambda}=I_H$. Then, both $\\Lambda$ and $\\Theta$ are g-fusion frames.\n\\end{theorem}\n\\begin{proof}\nFor each $f\\in H$ we have\n\\begin{align*}\n\\Vert f\\Vert^4&=\\langle f, f\\rangle^2\\\\\n&=\\langle T^*_{\\Lambda}f, T^*_{\\Theta}f\\rangle^2\\\\\n&\\leq\\Vert T^*_{\\Lambda}f\\Vert^2 \\Vert T^*_{\\Theta}f\\Vert^2\\\\\n&=\\big(\\sum_{j\\in\\Bbb I}v_j^2\\Vert \\Lambda_j \\pi_{W_j}f\\Vert^2\\big)\\big(\\sum_{j\\in\\Bbb I}v_j^2\\Vert \\Theta_j \\pi_{W_j}f\\Vert^2\\big)\\\\\n&\\leq\\big(\\sum_{j\\in\\Bbb I}v_j^2\\Vert \\Lambda_j \\pi_{W_j}f\\Vert^2\\big) B_2 \\Vert f\\Vert^2,\n\\end{align*}\nthus, $B_2^{-1}\\Vert f\\Vert^2\\leq\\sum_{j\\in\\Bbb I}v_j^2\\Vert \\Lambda_j \\pi_{W_j}f\\Vert^2$. This means that $\\Lambda$ is a g-fusion frame for $H$. Similarly, $\\Theta$ is a g-fusion frame with the lower bound $B_1^{-1}$.\n\\end{proof}\n\\section{Dual g-Fusion Frames}\nFor definition of the dual g-fusion frames, we need the following theorem.\n\\begin{theorem}\\label{dual}\nLet $\\Lambda=(W_j, \\Lambda_j, v_j)$ be a g-fusion frame for $H$. Then $(S^{-1}_{\\Lambda}W_j, \\Lambda_j \\pi_{W_j}S_{\\Lambda}^{-1}, v_j)$ is a g-fusion frame for $H$.\n\\end{theorem}\n\\begin{proof}\n Let $A,B$ be the g-fusion frame bounds of $\\Lambda$ and $f\\in H$, then\n\\begin{align*}\n\\sum_{j\\in\\Bbb J}v^2_j\\Vert \\Lambda_{j}\\pi_{W_j}S_{\\Lambda}^{-1}\\pi_{S^{-1}_{\\Lambda}W_j}f\\Vert^2&=\\sum_{j\\in\\Bbb J}v^2_j\\Vert \\Lambda_{j}\\pi_{W_j}S_{\\Lambda}^{-1}f\\Vert^2\\\\\n&\\leq B\\Vert S_{\\Lambda}^{-1}\\Vert^2 \\Vert f\\Vert^2.\n\\end{align*}\nNow, to get the lower bound, by using (\\ref{3}) we can write\n\\begin{align*}\n\\Vert f\\Vert^4&=\\big\\vert\\langle\\sum_{j\\in\\Bbb J}v_j^2\\pi_{W_j}\\Lambda^*_J\\Lambda_j\\pi_{W_j}S^{-1}_{\\Lambda}f, f\\rangle\\big\\vert^2\\\\\n&=\\big\\vert\\sum_{j\\in\\Bbb J}v_j^2\\langle\\Lambda_j\\pi_{W_j}S^{-1}_{\\Lambda}f, \\Lambda_j\\pi_{W_j}f\\rangle\\big\\vert^2\\\\\n&\\leq\\sum_{j\\in\\Bbb J}v_j^2\\Vert \\Lambda_j\\pi_{W_j}S^{-1}_{\\Lambda}f\\Vert^2 \\sum_{j\\in\\Bbb J}v_j^2\\Vert\\Lambda_j\\pi_{W_j}f\\Vert^2\\\\\n&\\leq \\sum_{j\\in\\Bbb J}v_j^2\\Vert\\Lambda_j\\pi_{W_j} S_{\\Lambda}^{-1}\\pi_{S^{-1}_{\\Lambda}W_j}f\\Vert^2\\big(B\\Vert f\\Vert^2\\big),\n\\end{align*}\ntherefore\n\\begin{align*}\nB^{-1}\\Vert f\\Vert^2\\leq \\sum_{j\\in\\Bbb J}v_j^2\\Vert\\Lambda_j\\pi_{W_j} S_{\\Lambda}^{-1}\\pi_{S^{-1}_{\\Lambda}W_j}f\\Vert^2.\n\\end{align*}\n\\end{proof}\nNow, by Theorem \\ref{dual}, $\\tilde{\\Lambda}=(S^{-1}_{\\Lambda}W_j, \\Lambda_j\\pi_{W_j} S_{\\Lambda}^{-1}, v_j)$ is a g-fusion frame for $H$. Then, $\\tilde{\\Lambda}$ is called the \\textit{(canonical) dual g-fusion frame} of $\\Lambda$. Let $S_{\\tilde{\\Lambda}}=T_{\\tilde{\\Lambda}}T^*_{\\tilde{\\Lambda}}$ is the g-fusion frame operator of $\\tilde{\\Lambda}$. Then, for each $f\\in H$ we get\n$$T^*_{\\tilde{\\Lambda}}f=\\lbrace v_j\\Lambda_j\\pi_{W_j}S^{-1}_{\\Lambda}\\pi_{S^{-1}_{\\Lambda W_j}}f\\rbrace=\\lbrace v_j\\Lambda_j\\pi_{W_j} S^{-1}_{\\Lambda}f\\rbrace=T^*_{\\Lambda}(S^{-1}_{\\Lambda}f),$$\nso $T_{\\Lambda}T^*_{\\tilde{\\Lambda}}=I_H$. Also, we have for each $f\\in H$,\n\\begin{align*}\n\\langle S_{\\tilde{\\Lambda}}f, f\\rangle&=\\sum_{j\\in\\Bbb J}v_j^2\\Vert\\Lambda_j\\pi_{W_j} S_{\\Lambda}^{-1}\\pi_{S^{-1}_{\\Lambda}W_j}f\\Vert^2\\\\\n&=\\sum_{j\\in\\Bbb J}v_j^2\\Vert\\Lambda_j \\pi_{W_j}S_{\\Lambda}^{-1}f\\Vert^2\\\\\n&=\\langle S_{\\Lambda}(S_{\\Lambda}^{-1}f), S_{\\Lambda}^{-1}f\\rangle\\\\\n&=\\langle S_{\\Lambda}^{-1}f, f\\rangle\n\\end{align*}\nthus, $S_{\\tilde{\\Lambda}}=S_{\\Lambda}^{-1}$ and by (\\ref{3}), we get for each $f\\in H$\n\\begin{align}\\label{frame}\nf=\\sum_{j\\in\\Bbb J}v_j^2\\pi_{W_j}\\Lambda^*_j\\tilde{\\Lambda_j}\\pi_{\\tilde{W_j}}f=\n\\sum_{j\\in\\Bbb J}v_j^2\\pi_{\\tilde{W_j}}\\tilde{\\Lambda_j}^*\\Lambda_j\\pi_{W_j}f,\n\\end{align}\nwhere $\\tilde{W_j}:=S^{-1}_{\\Lambda}W_j \\ , \\ \\tilde{\\Lambda_j}:=\\Lambda_j \\pi_{W_j}S_{\\Lambda}^{-1}.$\n\nThe following Theorem shows that the canonical dual g-fusion frame\ngives rise to expansion coefficients with the minimal norm.\n\\begin{theorem}\\label{min}\nLet $\\Lambda$ be a g-fusion frame with canonical dual $\\tilde{\\Lambda}$. \nFor each $g_j\\in H_j$, put $f=\\sum_{j\\in\\Bbb J}v_j^2\\pi_{W_j}\\Lambda^*_j g_j$. Then\n$$\\sum_{j\\in\\Bbb J}\\Vert g_j\\Vert^2=\\sum_{j\\in\\Bbb J}v_j^2\\Vert \\tilde{\\Lambda_j}\\pi_{\\tilde{W_j}}f\\Vert^2+\\sum_{j\\in\\Bbb J}\\Vert g_j-v_j^2\\tilde{\\Lambda_j}\\pi_{\\tilde{W_j}}f\\Vert^2.$$\n\\end{theorem}\n\\begin{proof}\nWe can write again\n\\begin{align*}\n\\sum_{j\\in\\Bbb J}v_j^2\\Vert \\tilde{\\Lambda_j}\\pi_{\\tilde{W_j}}f\\Vert^2&=\\langle f, S^{-1}_{\\Lambda}f\\rangle\\\\\n&=\\sum_{j\\in\\Bbb J}v_j^2\\langle\\pi_{W_j}\\Lambda^*_j g_j, S_{\\Lambda}^{-1}f \\rangle\\\\\n&=\\sum_{j\\in\\Bbb J}v_j^2\\langle g_j, \\Lambda_j\\pi_{W_j}S_{\\Lambda}^{-1}f \\rangle\\\\\n&=\\sum_{j\\in\\Bbb J}v_j^2\\langle g_j, \\tilde{\\Lambda_j}\\pi_{\\tilde{W_j}} f \\rangle.\n\\end{align*}\nTherefore, $\\mbox{Im}\\Big(\\sum_{j\\in\\Bbb J}v_j^2\\langle g_j, \\tilde{\\Lambda_j}\\pi_{\\tilde{W_j}} f \\rangle\\Big)=0$. So\n\\begin{align*}\n\\sum_{j\\in\\Bbb J}\\Vert g_j-v_j^2\\tilde{\\Lambda_j}\\pi_{\\tilde{W_j}}f\\Vert^2=\\sum_{j\\in\\Bbb J}\\Vert g_j\\Vert^2 -2\\sum_{j\\in\\Bbb J}v_j^2\\langle g_j, \\tilde{\\Lambda_j}\\pi_{\\tilde{W_j}} f \\rangle+\\sum_{j\\in\\Bbb J}v_j^2\\Vert \\tilde{\\Lambda_j}\\pi_{\\tilde{W_j}}f\\Vert^2\n\\end{align*}\nand the proof completes.\n\\end{proof}\n\\section{Gf-Complete and g-Fusion Frame Sequences}\n\\begin{definition}\nWe say that $(W_j, \\Lambda_j)$ is \\textit{gf-complete} , if\n$\\overline{\\mbox{span}}\\lbrace \\pi_{W_j}\\Lambda^*_j H_j\\rbrace=H.$\n\\end{definition}\nNow, it is easy to check that $(W_j, \\Lambda_j)$ is gf-complete if and only if\n$$\\lbrace f: \\ \\Lambda_j \\pi_{W_j}f=0 , \\ j\\in\\Bbb J\\rbrace=\\lbrace 0\\rbrace.$$\n\\begin{proposition}\\label{p3}\nIf $\\Lambda=(W_j, \\Lambda_j, v_j)$ is a g-fusion frame for $H$, then $(W_j, \\Lambda_j)$ is a gf-complete.\n\\end{proposition}\n\\begin{proof}\nLet $f\\in(\\mbox{span}\\lbrace \\pi_{W_j}\\Lambda^*_j H_j\\rbrace)^{\\perp}\\subseteq H$. For each $j\\in\\Bbb J$ and $g_j\\in H_j$ we have\n$$\\langle \\Lambda_j\\pi_{W_j}f, g_j\\rangle=\\langle f, \\pi_{W_j}\\Lambda^*_j g_j\\rangle=0,$$\nso, $\\Lambda_j\\pi_{W_j}f=0$ for all $j\\in\\Bbb J$. Since $\\Lambda$ is a g-fusion frame for $H$, then $\\Vert f\\Vert=0$. Thus $f=0$ and we get $(\\mbox{span}\\lbrace \\pi_{W_j}\\Lambda^*_j H_j\\rbrace)^{\\perp}=\\lbrace0\\rbrace$.\n\\end{proof}\nIn the following, we want to check that if a member is removed from a g-fusion frame, will the new set remain a g-fusion frame or not?\n\\begin{theorem}\\label{del}\nLet $\\Lambda=(W_j, \\Lambda_j, v_j)$ be a g-fusion frame for $H$ with bounds $A, B$ and $\\tilde{\\Lambda}=(S^{-1}_{\\Lambda}W_j, \\Lambda_j \\pi_{W_j}S^{-1}_{\\Lambda}, v_j)$ be a canonical dual g-fusion frame. Suppose that $j_0\\in\\Bbb J$.\n\\begin{enumerate}\n\\item If there is a $g_0\\in H_{j_0}\\setminus\\lbrace 0\\rbrace$ such that $\\tilde\\Lambda_{j_0}\\pi_{\\tilde{W}_{j_0}}\\pi_{W_{j_0}}\\Lambda^*_{j_0}g_0=g_0$ and $v_{j_0}=1$, then $(W_j, \\Lambda_j)_{j\\neq j_0}$ is not gf-complete in $H$.\n\\item If there is a $f_0\\in H_{j_0}\\setminus\\lbrace0\\rbrace$ such that $\\pi_{W_{j_0}}\\Lambda^*_{j_0}\\tilde\\Lambda_{j_0}\\pi_{\\tilde{W}_{j_0}}f_0=f_0$ and $v_{j_0}=1$, then $(W_j, \\Lambda_j)_{j\\neq j_0}$ is not gf-complete in $H$.\n\\item If $I-\\Lambda_{j_0}\\pi_{W_{j_0}}\\pi_{\\tilde{W_{j_0}}}\\tilde{\\Lambda}^*_{j_0}$ is bounded invertible on $H_{j_0}$, then $(W_j, \\Lambda_j, v_j)_{j\\neq j_0}$ is a g-fusion frame for $H$.\n\\end{enumerate}\n\\end{theorem}\n\\begin{proof}\n\\textit{(1).} Since $\\pi_{W_{j_0}}\\Lambda^*_{j_0}g_0\\in H$, then by (\\ref{frame}),\n$$\\pi_{W_{j_0}}\\Lambda^*_{j_0}g_0=\\sum_{j\\in\\Bbb J}v_j^2\\pi_{W_j}\\Lambda^*_j\\tilde{\\Lambda_j}\\pi_{\\tilde{W_j}}\\pi_{W_{j_0}}\\Lambda^*_{j_0}g_0.$$\nSo,\n$$\\sum_{j\\neq j_0}v_j^2\\pi_{W_j}\\Lambda^*_j\\tilde{\\Lambda_j}\\pi_{\\tilde{W_j}}\\pi_{W_{j_0}}\\Lambda^*_{j_0}g_0=0.$$\nLet $u_{j_0, j}:=\\delta_{j_0, j}g_0$, thus\n$$\\pi_{W_{j_0}}\\Lambda^*_{j_0}g_0=\\sum_{j\\in\\Bbb J}v_j^2\\pi_{W_j}\\Lambda^*_j u_{j_0, j}.$$\nThen, by Theorem \\ref{min}, we have\n$$\\sum_{j\\in\\Bbb J}\\Vert u_{j_0, j}\\Vert^2=\\sum_{j\\in\\Bbb J}v_j^2\\Vert \\tilde{\\Lambda}_j\\pi_{\\tilde{W_j}}\\pi_{W_{j_0}}\\Lambda^*_{j_0}g_0\\Vert^2+\\sum_{j\\in\\Bbb J}\\Vert v_j^2\\tilde{\\Lambda}_j\\pi_{\\tilde{W_j}}\\pi_{W_{j_0}}\\Lambda^*_{j_0}g_0-u_{j_0, j}\\Vert^2.$$\nConsequently,\n$$\\Vert g_0\\Vert^2=\\Vert g_0\\Vert^2+2\\sum_{j\\neq j_0}v_j^2\\Vert \\tilde{\\Lambda}_j\\pi_{\\tilde{W_j}}\\pi_{W_{j_0}}\\Lambda^*_{j_0}g_0\\Vert^2$$\nand we get $\\tilde{\\Lambda}_j\\pi_{\\tilde{W_j}}\\pi_{W_{j_0}}\\Lambda^*_{j_0}g_0=0$.\nTherefore,\n$$\\Lambda_j\\pi_{W_j}S^{-1}_{\\Lambda}\\pi_{W_{j_0}}\\Lambda^*_{j_0}g_0=\\tilde{\\Lambda}_j\\pi_{\\tilde{W_j}}\\pi_{W_{j_0}}\\Lambda^*_{j_0}g_0=0.$$\nBut, $g_0=\\tilde\\Lambda_{j_0}^*\\pi_{\\tilde{W}_{j_0}}\\pi_{W_{j_0}}\\Lambda^*_{j_0}g_0=\\Lambda_{j_0}\\pi_{W_{j_0}}S^{-1}_{\\Lambda}\\pi_{W_{j_0}}\\Lambda^*_{j_0}g_0\\neq0$, which implies that $S^{-1}_{\\Lambda}\\pi_{W_{j_0}}\\Lambda^*_{j_0}g_0\\neq0$ and this means that $(W_j, \\Lambda_j)_{j\\neq j_0}$ is not gf-complete in $H$.\n\n\\textit{(2).} Since $\\pi_{W_{j_0}}\\Lambda^*_{j_0}\\tilde\\Lambda_{j_0}\\pi_{\\tilde{W}_{j_0}}f_0=f_0\\neq0$, we obtain $\\tilde\\Lambda_{j_0}\\pi_{\\tilde{W}_{j_0}}f_0\\neq0$ and\n$$\\tilde\\Lambda_{j_0}\\pi_{\\tilde{W}_{j_0}}\\pi_{W_{j_0}}\\Lambda^*_{j_0}\\tilde\\Lambda_{j_0}\\pi_{\\tilde{W}_{j_0}}f_0=\\tilde\\Lambda_{j_0}\\pi_{\\tilde{W}_{j_0}}f_0.$$\nNow, the conclusion follows from \\textit{(1)}.\n\n\\textit{(3)}. Using (\\ref{frame}), we have for any $f\\in H$\n$$\\Lambda_{j_0}\\pi_{W_{j_0}}f=\\sum_{j\\in\\Bbb J}v_j^2\\Lambda_{j_0}\\pi_{W_{j_0}}\\pi_{\\tilde{W_j}}\\tilde{\\Lambda}^*_j\\Lambda_j\\pi_{W_j}f.$$\nSo,\n\\begin{equation}\\label{com}(I-\\Lambda_{j_0}\\pi_{W_{j_0}}\\pi_{\\tilde{W_{j_0}}}\\tilde{\\Lambda}^*_{j_0})\\Lambda_{j_0}\\pi_{W_{j_0}}f=\\sum_{j\\neq j_0}v_j^2\\Lambda_{j_0}\\pi_{W_{j_0}}\\pi_{\\tilde{W_j}}\\tilde{\\Lambda}^*_j\\Lambda_j\\pi_{W_j}f.\n\\end{equation}\nOn the other hand, we can write\n\\begin{small}\n\\begin{align*}\n\\big\\Vert\\sum_{j\\neq j_0}v_j^2\\Lambda_{j_0}\\pi_{W_{j_0}}\\pi_{\\tilde{W_j}}\\tilde{\\Lambda}^*_j\\Lambda_j\\pi_{W_j}f\\big\\Vert^2&=\\sup_{\\Vert g\\Vert=1}\\big\\vert\\sum_{j\\neq j_0}v_j^2\\big\\langle \\Lambda_j\\pi_{W_j}f, \\tilde{\\Lambda}_j\\pi_{\\tilde{W}_j}\\pi_{W_{j_0}}\\Lambda^*_{j_0}g \\big\\rangle\\big\\vert^2\\\\\n&\\leq\\big(\\sum_{j\\neq j_0}v_j^2\\Vert \\Lambda_j\\pi_{W_j}f\\Vert^2\\big)\\sup_{\\Vert g\\Vert=1}\\sum_{j\\in\\Bbb J}v_j^2\\Vert \\tilde{\\Lambda}_j\\pi_{\\tilde{W}_j}\\pi_{W_{j_0}}\\Lambda^*_{j_0}g\\Vert^2\\\\\n&\\leq\\tilde{B}\\Vert\\Lambda_{j_0}\\Vert^2(\\sum_{j\\neq j_0}v_j^2\\Vert \\Lambda_j\\pi_{W_j}f\\Vert^2)\n\\end{align*}\n\\end{small}\nwhere, $\\tilde{B}$ is the upper bound of $\\tilde\\Lambda$. Now, by (\\ref{com}), we have\n\\begin{equation*}\n\\Vert\\Lambda_{j_0}\\pi_{W_{j_0}}f\\Vert^2\\leq\\Vert(I-\\Lambda_{j_0}\\pi_{W_{j_0}}\\pi_{\\tilde{W_{j_0}}}\\tilde{\\Lambda}^*_{j_0})^{-1}\\Vert^2 \\tilde{B}\\Vert\\Lambda_{j_0}\\Vert^2(\\sum_{j\\neq j_0}v_j^2\\Vert \\Lambda_j\\pi_{W_j}f\\Vert^2).\n\\end{equation*}\nTherefore, there is a number $C>0$ such that\n$$\\sum_{j\\in\\Bbb J}v_j^2\\Vert \\Lambda_j\\pi_{W_j}f\\Vert^2\\leq C\\sum_{j\\neq j_0}v_j^2\\Vert \\Lambda_j\\pi_{W_j}f\\Vert^2$$\nand we conclude for each $f\\in H$\n$$\\frac{A}{C}\\Vert f\\Vert^2\\leq\\sum_{j\\neq j_0}v_j^2\\Vert \\Lambda_j\\pi_{W_j}f\\Vert^2\\leq B\\Vert f\\Vert^2.$$\n\\end{proof}\n\\begin{theorem}\n$\\Lambda$ is a g-fusion frame for $H$ with bounds $A,B$ if and only if the following two conditions are satisfied:\n\\begin{enumerate}\n\\item[(I)] The pair $(W_j, \\Lambda_j)$ is gf-complete.\n\\item[(II)] The operator\n$$T_{\\Lambda}: \\lbrace f_j\\rbrace_{j\\in\\Bbb J}\\mapsto \\sum_{j\\in\\Bbb J}v_j\\pi_{W_j}\\Lambda_j^* f_j$$\nis a well-defined from $\\mathscr{H}_2$ into $H$ and for each $\\lbrace f_j\\rbrace_{j\\in\\Bbb J}\\in\\mathcal{N}^{\\perp}_{T_{\\Lambda}}$,\n\\begin{equation}\\label{e7}\nA\\sum_{j\\in\\Bbb J}\\Vert f_j\\Vert^2\\leq \\Vert T_{\\Lambda}\\lbrace f_j\\rbrace_{j\\in\\Bbb J}\\Vert^2\\leq B\\sum_{j\\in\\Bbb J}\\Vert f_j\\Vert^2.\n\\end{equation}\n\\end{enumerate}\n\\end{theorem}\n\\begin{proof}\nFirst, suppose that $\\Lambda$ is a g-fusion frame. By Proposition \\ref{p3}, (I) is satisfied. By Theorem \\ref{t2}, $T_{\\Lambda}$ is a well-defined from $\\mathscr{H}_2$ into $H$ and $\\Vert T_{\\Lambda}\\Vert^2\\leq B$. Now, the right-hand inequality in (\\ref{e7}) is proved.\n\nBy Theorem \\ref{2.3}, $T_{\\Lambda}$ is surjective. So, $\\mathcal{R}_{T^*_{\\Lambda}}$ is closed. Thus\n$$\\mathcal{N}^{\\perp}_{T_{\\Lambda}}=\\overline{\\mathcal{R}_{T^*_{\\Lambda}}}=\\mathcal{R}_{T^*_{\\Lambda}}.$$\nNow, if $\\lbrace f_j\\rbrace_{j\\in\\Bbb J}\\in\\mathcal{N}^{\\perp}_{T_{\\Lambda}}$, then\n$$\\lbrace f_j\\rbrace_{j\\in\\Bbb J}=T^*_{\\Lambda}g=\\lbrace v_j\\Lambda_j \\pi_{W_j}g\\rbrace_{j\\in\\Bbb J}$$\nfor some $g\\in H$. Therefore\n\\begin{align*}\n(\\sum_{j\\in\\Bbb J}\\Vert f_j\\Vert^2)^2&=(\\sum_{j\\in\\Bbb J}v^2_j\\Vert \\Lambda_j \\pi_{W_j}g\\Vert^2)^2\n\\vert\\langle S_{\\Lambda}(g), g\\rangle\\vert^2\\\\\n&\\leq\\Vert S_{\\Lambda}(g)\\Vert^2 \\Vert g\\Vert^2\\\\\n&\\leq\\Vert S_{\\Lambda}(g)\\Vert^2 \\big(\\frac{1}{A}\\sum_{j\\in\\Bbb J}v^2_j\\Vert \\Lambda_j \\pi_{W_j}g\\Vert^2\\big).\n\\end{align*}\nThis implies that\n$$A\\sum_{j\\in\\Bbb J}\\Vert f_j\\Vert^2\\leq\\Vert S_{\\Lambda}(g)\\Vert^2=\\Vert T_{\\Lambda}\\lbrace f_j\\rbrace_{j\\in\\Bbb J}\\Vert^2$$\nand (II) is proved.\n\nConversely, Let $(W_j, \\Lambda_j)$ be gf-complete and inequality\n(\\ref{e7}) is satisfied. Let $\\lbrace t_j\\rbrace_{j\\in\\Bbb\nJ}=\\lbrace f_j\\rbrace_{j\\in\\Bbb J}+\\lbrace g_j\\rbrace_{j\\in\\Bbb J}$,\nwhere $\\lbrace f_j\\rbrace_{j\\in\\Bbb J}\\in\\mathcal{N}_{T_{\\Lambda}}$\nand $\\lbrace g_j\\rbrace_{j\\in\\Bbb\nJ}\\in\\mathcal{N}_{T_{\\Lambda}}^{\\perp}$. We get\n\\begin{align*}\n\\Vert T_{\\Lambda}\\lbrace t_j\\rbrace_{j\\in\\Bbb J}\\Vert^2&=\\Vert T_{\\Lambda}\\lbrace g_j\\rbrace_{j\\in\\Bbb J}\\Vert^2\\\\\n&\\leq B\\sum_{j\\in\\Bbb J}\\Vert g_j\\Vert^2\\\\\n&\\leq B\\Vert \\lbrace f_j\\rbrace+\\lbrace g_j\\rbrace\\Vert^2\\\\\n&=B\\Vert\\lbrace t_j\\rbrace_{j\\in\\Bbb J}\\Vert^2.\n\\end{align*}\nThus, $\\Lambda$ is a g-fusion Bessel sequence.\n\nAssume that $\\lbrace y_n\\rbrace$ is a sequence of members of $\\mathcal{R}_{T_{\\Lambda}}$ such that $y_n\\rightarrow y$ for some $y\\in H$. So, there is a $\\lbrace x_n\\rbrace\\in\\mathcal{N}_{T_{\\Lambda}}$ such that $T_{\\Lambda}\\{x_n\\}=y_n$. By (\\ref{e7}), we obtain\n\\begin{align*}\nA\\Vert\\lbrace x_n-x_m\\rbrace\\Vert^2&\\leq\\Vert T_{\\Lambda}\\lbrace x_n-x_m\\rbrace\\Vert^2\\\\\n&=\\Vert T_{\\Lambda}\\lbrace x_n\\rbrace -T_{\\Lambda}\\lbrace x_m\\rbrace\\Vert^2\\\\\n&=\\Vert y_n-y_m\\Vert^2.\n\\end{align*}\nTherefore, $\\lbrace x_n\\rbrace$ is a Cauchy sequence in $\\mathscr{H}_2$. Therefore $\\lbrace x_n\\rbrace$ converges to some $x\\in \\mathscr{H}_2$, which by continuity of $T_{\\Lambda}$, we have $y=T_{\\Lambda}(x)\\in\\mathcal{R}_{T_{\\Lambda_j}}$. Hence $\\mathcal{R}_{T_{\\Lambda}}$ is closed. Since $\\mbox{span}\\lbrace \\pi_{W_j}\\Lambda_{j}^{\\ast}(H_j)\\rbrace\\subseteq\\mathcal{R}_{T_{\\Lambda}}$, by (I) we get $\\mathcal{R}_{T_{\\Lambda}}=H$.\n\n Let $T_{\\Lambda}^\\dagger$ denotes the pseudo-inverse of $T_{\\Lambda}$. By Lemma \\ref{Ru}(4), $T_{\\Lambda}T_{\\Lambda}^{\\dagger}$ is the orthogonal projection onto $\\mathcal{R}_{T_{\\Lambda}}=H$. Thus for any $\\lbrace f_j\\rbrace_{j\\in\\Bbb J}\\in\\mathscr{H}_2 $,\n\\begin{eqnarray*}\nA\\Vert T_{\\Lambda}^{\\dagger}T_{\\Lambda}\\lbrace f_j\\rbrace\\Vert^2\\leq\\Vert T_{\\Lambda}T_{\\Lambda}^{\\dagger}T_{\\Lambda}\\lbrace f_j\\rbrace \\Vert^2=\\Vert T_{\\Lambda}\\lbrace f_j\\rbrace\\Vert^2.\n\\end{eqnarray*}\nBy Lemma \\ref{Ru} (4), $\\mathcal{N}_{{T}_{\\Lambda}^{\\dagger}}=\\mathcal{R}^{\\bot}_{T_{\\Lambda}}$, therefore\n\\begin{eqnarray*}\n\\Vert T_{\\Lambda}^\\dagger\\Vert^2\\leq\\frac{1}{A}.\n\\end{eqnarray*}\nAlso by Lemma \\ref{Ru}(2), we have\n$$ \\Vert(T_{\\Lambda}^\\ast)^{\\dagger}\\Vert^2\\leq\\frac{1}{A}.$$\nBut $(T_{\\Lambda}^\\ast)^{\\dagger}T_{\\Lambda}^\\ast$ is the\northogonal projection onto\n\\begin{eqnarray*}\n\\mathcal{R}_{(T_{\\Lambda}^\\ast)^\\dagger}=\\mathcal{R}_{(T_{\\Lambda}^\\dagger)^\\ast}=\\mathcal{N}_{T_{\\Lambda}^\\dagger}^{\\bot}=\\mathcal{R}_{T_{\\Lambda}}=H.\n\\end{eqnarray*}\nSo, for all $f\\in H$\n\\begin{align*}\n\\Vert f\\Vert^2&=\\Vert(T_{\\Lambda}^\\ast)^{\\dagger}T_{\\Lambda}^\\ast f\\Vert^2\\\\\n&\\leq \\frac{1}{A}\\Vert T_{\\Lambda}^\\ast f\\Vert^2\\\\\n&=\\frac{1}{A}\\sum_{j\\in\\Bbb J}v^2_j\\Vert \\Lambda_j \\pi_{W_j}f\\Vert^2.\n\\end{align*}\nThis implies that $\\Lambda$ satisfies the lower g-fusion frame condition.\n\\end{proof}\nNow, we can define a g-fusion frame sequence in the Hilbert space.\n\\begin{definition}\nWe say that $\\Lambda$ is a \\textit{g-fusion frame sequence} if it is a g-fusion frame for $\\overline{\\mbox{span}}\\lbrace \\pi_{W_j}\\Lambda^*_j H_j\\rbrace$.\n\\end{definition}\n\\begin{theorem}\\label{2.6}\n$\\Lambda$ is a g-fusion frame sequence if and only if the operator\n \\begin{align*}\nT_{\\Lambda}&:\\mathscr{H}_2\\longrightarrow H\\\\\nT_{\\Lambda}(\\lbrace f_j\\rbrace_{j\\in\\Bbb J})&=\\sum_{j\\in\\Bbb J}v_j \\pi_{W_j}\\Lambda_{j}^{*}f_j\n\\end{align*}\nis a well-defined and has closed range.\n\\end{theorem}\n\\begin{proof}\nBy Theorem \\ref{2.3}, it is enough to prove that if $T_{\\lambda}$ has closed range, then $\\overline{\\mbox{span}}\\lbrace \\pi_{W_j}\\Lambda^*_j H_j\\rbrace=\\mathcal{R}_{T_{\\Lambda}}$.\nLet $f\\in\\overline{\\mbox{span}}\\lbrace \\pi_{W_j}\\Lambda^*_j H_j\\rbrace$, then\n$$f=\\lim_{n\\rightarrow\\infty}g_n , \\ \\ \\ g_n\\in\\mbox{span}\\lbrace \\pi_{W_j}\\Lambda^*_j H_j\\rbrace\\subseteq \\mathcal{R}_{T_{\\Lambda}}=\\overline{\\mathcal{R}}_{T_{\\Lambda}}.$$\nTherefore, $\\overline{\\mbox{span}}\\lbrace \\pi_{W_j}\\Lambda^*_j H_j\\rbrace\\subseteq\\overline{\\mathcal{R}}_{T_{\\Lambda}}=\\mathcal{R}_{T_{\\Lambda}}$. On the other hand, if $f\\in\\mathcal{R}_{T_{\\Lambda}}$, then\n$$f\\in\\mbox{span}\\lbrace \\pi_{W_j}\\Lambda^*_j H_j\\rbrace\\subseteq\\overline{\\mbox{span}}\\lbrace \\pi_{W_j}\\Lambda^*_j H_j\\rbrace$$\n and the proof is completed.\n\\end{proof}\n\\begin{theorem}\n $\\Lambda$ is a g-fusion frame sequence if and only if\n\\begin{equation}\\label{4}\nf \\longmapsto \\lbrace v_j \\Lambda_j \\pi_{W_j}f\\rbrace_{j\\in\\Bbb J}\n\\end{equation}\ndefines a map from $H$ onto a closed subspace of $\\mathscr{H}_2$.\n\\end{theorem}\n\\begin{proof}\nLet $\\Lambda$ be a g-fusion frame sequence. Then, by Theorem \\ref{2.6}, $T_{\\lambda}$ is well-defined and $\\mathcal{R}_{T_{\\Lambda}}$ is closed. So, $T^*_{\\Lambda}$ is well-defined and has closed range. Conversely, by hypothesis, for all $f\\in H$\n$$\\sum_{j\\in\\Bbb J}\\Vert v_j \\Lambda_j \\pi_{W_j}f\\Vert^2<\\infty.$$\nLet\n$$B:=\\sup\\big\\lbrace \\sum_{j\\in\\Bbb J}v_j^2\\Vert \\Lambda_j \\pi_{W_j}f\\Vert^2 : \\ \\ f\\in H, \\ \\Vert f\\Vert=1\\big\\rbrace$$\nand suppose that $g_j\\in H_j$ and $\\Bbb I\\subseteq\\Bbb J$ be finite. We can write\n\\begin{align*}\n\\Vert\\sum_{j\\in\\Bbb I}v_j \\pi_{W_j}\\Lambda^*_j g_j\\Vert^2&=\\Big(\\sup_{\\Vert f\\Vert=1}\\big\\vert\\langle\\sum_{j\\in\\Bbb I}v_j \\pi_{W_j}\\Lambda^*_j g_j, f\\rangle\\big\\vert\\Big)^2\\\\\n&\\leq\\Big(\\sup_{\\Vert f\\Vert=1}\\sum_{j\\in\\Bbb I}v_j\\big\\vert\\langle g_j, \\Lambda_j \\pi_{W_j}f\\rangle\\big\\vert\\Big)^2\\\\\n&\\leq\\big(\\sum_{j\\in\\Bbb I}\\Vert g_j\\Vert^2\\big)\\big(\\sup_{\\Vert f\\Vert=1}\\sum_{j\\in\\Bbb I}v_j^2\\Vert \\Lambda_j \\pi_{W_j}f\\Vert^2\\big)\\\\\n&\\leq B\\big(\\sum_{j\\in\\Bbb I}\\Vert g_j\\Vert^2\\big)\n\\end{align*}\nThus, by Corollary \\ref{cor}, $\\Lambda$ is a g-fusion Bessel sequence for $H$. Therefore, $T_{\\Lambda}$ is well-defined and bounded. Furthermore, if the range of the map (\\ref{4}) is closed, the same is true for $T_{\\Lambda}$. So, by Theorem \\ref{2.6}, $\\Lambda$ is a g-fusion frame sequence.\n\\end{proof}\n\\begin{theorem}\nLet $\\Lambda=(W_j, \\Lambda_j, v_j)$ be a g-fusion frame sequence Then, it is a g-fusion frame for $H$ if and only if the map\n\\begin{equation}\\label{5}\nf \\longmapsto \\lbrace v_j \\Lambda_j \\pi_{W_j}f\\rbrace_{j\\in\\Bbb J}\n\\end{equation}\nfrom $H$ onto a closed subspace of $\\mathscr{H}_2$ be injective.\n\\end{theorem}\n\\begin{proof}\nSuppose that the map (\\ref{5}) is injective and $v_j \\Lambda_j \\pi_{W_j}f=0$ for all $j\\in\\Bbb J$. Then, the value of the map at $f$ is zero. So, $f=0$. This means that $(W_j, \\Lambda_j)$ is gf-complete. Since, $\\Lambda$ is a g-fusion frame sequence, so, it is a g-fusion frame for $H$.\n\nThe converse is evident.\n\\end{proof}\n\\begin{theorem}\nLet $\\Lambda$ be a g-fusion frame for $H$ and $u\\in\\mathcal{B}(H)$. Then $\\Gamma:=(uW_j, \\Lambda_j u^*, v_j)$ is a g-fusion frame sequence if and only if $u$ has closed range.\n\\end{theorem}\n\\begin{proof}\nAssume that $\\Gamma$ is a g-fusion frame sequence. So, by Theorem \\ref{2.6}, $T_{\\Lambda u^*}$ is a well-defined operator from $\\mathscr{H}_2$ into $H$ with closed range. If $\\lbrace f_j\\rbrace_{j\\in\\Bbb J}\\in\\mathscr{H}_2$, then\n\\begin{align*}\nuT_{\\Lambda}\\lbrace f_j\\rbrace_{j\\in\\Bbb J}&=\\sum_{j\\in\\Bbb J}v_ju\\pi_{W_j}\\Lambda_j^* f_j\\\\\n&=\\sum_{j\\in\\Bbb J}v_j\\pi_{uW_j}u\\Lambda_j^* f_j\\\\\n&=\\sum_{j\\in\\Bbb J}v_j\\pi_{uW_j}(\\Lambda_j u^*)^* f_j\\\\\n&=T_{\\Lambda u^*}\\lbrace f_j\\rbrace_{j\\in\\Bbb J},\n\\end{align*}\ntherefore $uT_{\\Lambda}=T_{\\Lambda u^*}$. Thus $uT_{\\Lambda}$ has closed range too. Let $y\\in\\mathcal{R}_u$, then there is $x\\in H$ such that $u(x)=y$. By Theorem \\ref{2.3}, $T_{\\Lambda}$ is surjective, so there exist $\\{f_j\\}_{j\\in\\Bbb J}\\in\\mathscr{H}_2$ such that \n$$y=u(T_{\\Lambda}\\{f_j\\}_{j\\in\\Bbb J}).$$\nThus, $\\mathcal{R}_{u}=\\mathcal{R}_{uT_{\\Lambda}}$ and $u$ has closed range.\n\nFor the opposite implication, let\n\\begin{align*}\nT_{\\Lambda u^*}:&\\mathscr{H}_2\\longrightarrow H\\\\\nT_{\\Lambda u^*}\\lbrace f_j\\rbrace_{j\\in\\Bbb J}&=\\sum_{j\\in\\Bbb J}v_j\\pi_{uW_j}(\\Lambda_j u^*)^* f_j.\n\\end{align*}\nHence, $T_{\\Lambda u^*}=uT_{\\Lambda}$. Since, $T_{\\Lambda}$ is surjective, so $T_{\\Lambda u^*}$ has closed range and by Theorem \\ref{t2}, is well-defined. Therefore, by Theorem \\ref{2.6}, the proof is completed.\n\\end{proof}\n\\section{Conclusions}\nIn this paper, we could transfer some common properties in general frames to g-fusion frames with the definition of the g-fusion frames and their operators. Afterward, we reviewed a basic theorem about deleting a member in Theorem \\ref{del} with the definition of the dual g-fusion frames and the gf-completeness. In this theorem, the defined operator in part \\textit{3} could be replaced by some other operators which are the same as the parts \\textit{1} and \\textit{2}; this is an open problem at the moment. Eventually, the g-fusion frame sequences and their relationship with the closed range operators were defined and presented.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n In 1992, the pioneering work of Allen and coworkers~\\cite{Allen1992} showed that twisted light beams also posses orbital angular momentum (OAM) in addition to spin angular momentum. Since then, research on twisted (or vortex) light beams has been a burgeoning research area in the scientific community \\cite{Yao,opticaloam1,opticaloam2}. The presence of extra angular momentum in the twisted light beam is associated with an azimuthally varying phase of the beam \\cite{padgett2004light}. In addition to OAM, twisted light beams exhibit helical phase fronts and phase singularity along the beam axis \\cite{twistedlight}. In contrast to the plane waves, the twisted light beams have a non-uniform intensity distribution across the beam cross section. In particular, we observe concentric rings in the beam cross section of the twisted light beams \\cite{molina2007twisted}. \n\\par Twisted light beams offer a plethora of applications because of their well-defined projection of the total angular momentum (TAM) upon the propagation axis. For example, twisted light beams have been utilized as alphabets to encode information beyond one bit per single photon \\cite{twistalphabets}. This makes twisted light beams such as Laguerre-Gaussian (LG) beams very suitable for a number of applications such as high-dimensional quantum information \\cite{highquantuminfo}, quantum memories \\cite{quantummemory}, quantum cryptography \\cite{quantumcrypto}. In addition, LG beams are also used in high harmonic generation \\cite{Willi} and optical tweezers \\cite{opticaltweezers}. \n\\par Atomic processes with twisted light beams, such as photo-ionization of atoms \\cite{photoionization1,photoionization2} and the scattering of twisted light beams by ions \\cite{rayleighscatt} or electrons \\cite{comptonscattering} have also gained much attention. It was shown that OAM of the twisted light beam can strongly modify the angular distribution of the emitted photo-electrons during the ionization process \\cite{angulardistribution}. In addition to photo-ionization and scattering processes, the photo-excitation of atoms by twisted light beams has attracted much interest in recent years. The transfer of OAM from the LG beams to the bound electrons in an atomic system during the excitation process was observed for the first time by Schmiegelow \\textit{et al.} \\cite{schmiegelow2016}. This experiment investigated the electric-quadrupole transition in the Ca$^{+}$ ion positioned on the beam axis and demonstrated the suppression of AC-Stark shift in the dark centre of the LG beams. Later, a theoretical study on the photo-excitation of atoms by twisted light beams showed that the magnetic sub-level population and fluorescence following the excitation process varied significantly with OAM of the LG beams \\cite{peshkov}.\n\\par While most studies on the photo-excitation of atoms have focused so far on circularly polarized LG beams, twisted light beams support many different polarization states, such as cylindrical polarization. However, photo-excitation of atoms by cylindrically polarized LG beams remains less explored. In this work, we therefore analyze the multipole distribution of the cylindrically polarized LG beams. We especially explore the dependence of the multipole distribution of cylindrically polarized LG beams on the beam waist and the radial distance from the beam axis. We show that the varying multipole distribution in the beam cross section influences the magnetic sub-level population of the target atom. As an example, we consider the electric-quadrupole transition $4s\\;^{2}S_{1\/2} \\rightarrow 3d\\;^{2}D_{5\/2}$ in the Ca$^{+}$ ion. This particular transition has already been observed in the past experimental as well as studied in theoretical works and thus serves as an ideal case for a comparison of our theoretical work. \n\\par This paper is structured as follows. In Sec. \\ref{subsec: Circularly polarized LG beams} we first recall circularly polarized LG beams. We then use the knowledge of circularly polarized LG beams to understand the cylindrical polarization and to construct the vector potential of cylindrically polarized LG beams in Sec. \\ref{subsec:Cylindrically polarized LG beams}. Moreover, we expand the vector potential of LG beam into its multipole components to obtain complex weight factors for a given polarization in Sec. \\ref{subsec:Multipole expansion of a vector potential of the LG beams}. We evaluate strength of the electric-quadrupole field in the beam cross section of a radially polarized LG beam with respect to circularly polarized LG beam in Sec. \\ref{subsec:Distribution of electric-quadrupole component in the beam cross section of cylindrically polarized LG beam}. Furthermore, in Sec. \\ref{subsec:Effect of the beam waist and radial position on the multipole distribution of LG beams} we analyze distribution of the projection of electric-quadrupole field in the beam cross section with respect to the beam waist $w_{o}$ and radial distance $b$ from the beam axis of cylindrically polarized LG beams. In the Sec. \\ref{subsec:Interaction of LG beams with the target atom}, we analyze the photo-excitation of the target Ca$^{+}$ ion by LG beams using the outcome of the multipole distribution and the selection rules. Finally, a summary of the paper is given in the Sec. \\ref{sec:summary}\n\n\\section{Theoretical background}\n\\subsection{Circularly polarized LG beams} \n\\label{subsec: Circularly polarized LG beams}\nLG beams are paraxial twisted light beams whose amplitude distribution $u(r)$ is known to satisfy the paraxial wave equation \\cite{siegman1986lasers}\n \\begin{equation}\n \\nabla^{2}\\;u\\;+2\\;i\\;k\\;\\frac{\\partial}{\\partial z}u \\; = \\; 0.\n \\end{equation}\nThis \\textit{paraxial wave} approximation is valid, if the amplitude distribution $u(\\bm{r})$ changes slowly with the distance $z$ and this $z$ dependence is less compared to variations of $u(\\bm{r})$ in the transverse direction \n\\begin{equation}\n |\\frac{\\partial^{2}u}{\\partial z^{2}}| \\ll |2k \\; \\frac{\\partial u}{\\partial z}| \\; , \\; |\\frac{\\partial^{2}u}{\\partial z^{2}}| \\ll |\\partial_{t}^{2}u|.\n\\end{equation}\nIn cylindrical coordinates the amplitude distribution of the LG beam is given by\n \\begin{widetext}\n \\begin{equation}\\label{eq:amplitude}\n u(\\bm{r}) = \\frac{1}{w(z)} \\left( \\frac{\\sqrt{2}r}{w(z)} \\right)^{m_{l}} \\textrm{exp}\\left(- \\frac{r^{2}}{w^{2}(z)} \\right) L^{m_{l}}_{p}\\left(\\frac{2r^{2}}{w^{2}(z)} \\right)\n \\textrm{exp}\\left[ im_{l}\\phi + \\frac{ikr^{2}z}{2(z^{2}+z^{2}_{R})} - i(2p+m_{l}+1) \\textrm{arctan}\\left(\\frac{z}{z_{R}}\\right) \\right],\n \\end{equation}\nwhere $m_{l}$ is the projection of the OAM upon the propagation axis, $p$ is the radial index of the LG beam, $z_{R}$ is the (so-called) Rayleigh range, $w(z)$ is the beam width of the LG beam. The beam width $w(z)$ of the LG beam varies along the propagation distance and is minimum for $z = 0$. This minimum beam width of a LG beam is known also as beam waist $w_{o}$ $\\equiv w(z = 0)$. Equation (\\ref{eq:amplitude}) can be used to obtain the intensity distribution $|u(r)|^{2}$ of the LG beam which exhibit a concentric ring-like structure in the beam cross section.\n\\par The wave amplitude of the circularly polarized LG beam in momentum space is expressed as a Fourier transformation of the amplitude distribution (\\ref{eq:amplitude}). Then the vector potential of a circularly polarized LG beam in Coulomb gauge is given by (see \\cite{peshkov} for a detailed derivation)\n \\begin{equation} \\label{eq:vectorpotential}\n \\mathbf{A}^{\\mathrm{cir}}_{m_{l},\\lambda,p}\\bm{(r)} = \\int d^{2}\\bm{k}_{\\bot} v_{pm_{l}}(k_{\\bot})\\; e^{i(m_{l}+\\lambda)\\phi_{k}}\\; \\mathbf{e}_{k,\\lambda} \\;e^{i \\mathbf{k}\\cdot \\mathbf{r}} ,\n\\end{equation}\nwhere $v_{pm_{l}}(k_{\\bot})\\; e^{i(m_{l}+\\lambda)\\phi_{k}}$ is the momentum space wave function and $\\mathbf{e}_{k,\\lambda} \\;e^{i \\mathbf{k}\\cdot \\mathbf{r}}$ is the vector potential of a circularly polarized plane waves. In the above equation momentum space wave function $v_{pm_{l}}(k_{\\bot})$ is given by\n \\begin{equation}\n v_{pm_{l}}(k_{\\bot}) = \\frac{(-i)^{m_{l}}}{ w_{o} 4\\pi} \\; e^{-k_{\\bot}^{2}w^{2}_{o}\/4} \\; \\left(\\frac{k_{\\bot} w_{o}}{2}\\right)^{m_{l}} \n \\sum^{p}_{\\beta=0} (-1)^{\\beta} \\; 2^{\\beta + m_{l}\/2} \\; \\left( p+m_{l} \\atop p-\\beta \\right) \\;L^{m_{l}}_{\\beta}\\left( \\frac{k_{\\bot}^{2}w^{2}_{o}}{4} \\right).\n\\end{equation}\n\\end{widetext}\nAs seen from equation (\\ref{eq:vectorpotential}), the LG beams with circular polarization can be expressed as a coherent superposition of circularly polarized plane waves in the momentum space. The momentum vector $\\bm{k}$ of these plane waves lie on the surface of a cone in the momentum space with an opening angle of $\\theta_{k}$ = arctan$(k_{\\bot}\/k_{z})$. \n\n\\subsection{Cylindrically polarized LG beams}\n\\label{subsec:Cylindrically polarized LG beams}\nCylindrically polarized LG beams can be constructed as a linear combination of two circularly polarized LG modes \\cite{cylindrica}. The superposition of two LG modes results in non-separable spatial and polarization modes \\cite{andrews2012} which affects the state of polarization across the beam cross section. In contrast to LG beams with circular polarization, the state of polarization across the beam cross section in cylindrically polarized LG beams is spatially in-homogeneous \\cite{nonuniformsop}. Beams which are linear combinations of two LG modes are known as vector beams \\cite{vectorsolution} and constitute a vector solutions to the paraxial wave equation. \n\\par Radial and azimuthal polarizations are two special cases of a cylindrical polarization. In particular, radially and azimuthally polarized LG beams are linear combinations of two LG modes with the projection of OAM $m_{l} = \\pm1$ and helicity $\\lambda = \\pm1$ \\cite{linearcombinationofcircular}. The radially polarized LG beams have a state of linear polarization aligned along the radial direction \\cite{radiallypol,sabrina}. That is, the electric field of the radially polarized LG beam always points in the radial direction and is perpendicular to the beam axis. The vector potential of a radially polarized LG beam is constructed as a linear combination of vector potential of right circularly polarized $\\mathbf{A}^{\\mathrm{cir}}_{m_{l}=-1,\\lambda=1,p}(\\mathbf{r})$ and left circularly polarized $\\mathbf{A}^{\\mathrm{cir}}_{m_{l}=1,\\lambda=-1,p}(\\mathbf{r})$ LG beams and is given by\n\\begin{equation}\n \\mathbf{A}^{\\mathrm{rad}}_{p}(\\mathbf{r}) = \\frac{-i}{\\sqrt{2}} \\left[ \\bm{A}^{\\mathrm{cir}}_{m_{l}=1,\\lambda =-1,p}(\\mathbf{r}) + \\bm{A}^{\\mathrm{cir}}_{m_{l}=-1,\\lambda =1,p}(\\mathbf{r}) \\right].\n\\end{equation}\n\n\\par For an azimuthally polarized LG beam, the state of linear polarization is always aligned tangential \\cite{azimuthal,sabrina} to the ring of the beam. The electric field direction of the azimuthally polarized LG beams is always perpendicular to the radial direction and its vector potential is given by\n \\begin{equation}\n \\mathbf{A}^{\\mathrm{azim}}_{p}(\\mathbf{r}) = \\frac{1}{\\sqrt{2}} \n \\left[ \\bm{A}^{\\mathrm{cir}}_{m_{l}=1,\\lambda=-1,p}(\\mathbf{r}) - \\bm{A}^{\\mathrm{cir}}_{m_{l}=-1,\\lambda=1,p}(\\mathbf{r}) \\right].\n\\end{equation}\nFurther, we use the vector potential of LG beam of a given polarization to obtain the complex weight factors using the multipole expansion in the next section.\n\n\\subsection{Multipole expansion of a vector potential of the LG beam }\n\\label{subsec:Multipole expansion of a vector potential of the LG beams}\nA multipole expansion of the radiation field enables us generally to expand the vector potential in angular momentum basis \\cite{brinkangularmomentum,rose1995elementary}. It helps us to analyze the contributions of the individual multipole components to the photo-excitation of the atoms. Since the intensity distribution of the LG beam is nonuniform in the beam cross section, we perform multipole expansion of the vector potential of a LG beam at a radial distance $b$ from the beam axis. That is, when the z axis is translated by a vector $\\bm{b} = b\\bm{e_{x}}$ from the beam axis.\nThe vector potential of a circularly polarized LG beam at a position $b$ from the beam axis is given by\n\\begin{equation}\n \\mathbf{A}^{\\mathrm{cir}}_{m_{l},\\lambda,p}(\\bm{r}; b,w_{o}) = \\mathbf{A}^{\\mathrm{cir}}_{m_{l},\\lambda,p}(\\bm{r}) \\; e^{i\\bm{p}\\cdot\\bm{b}}.\n\\end{equation}\nThe multipole expansion of the circularly polarized LG beam is given by\n\\begin{equation}\n \\small{\\mathbf{A}^{\\mathrm{cir}}_{m_{l},\\lambda,p}(\\bm{r};b,w_{o} ) \\; = \\; \\sum_{L,M,\\Lambda} W^{\\mathrm{cir}}_{m_{l},\\lambda,p}(L,M,\\Lambda; b,w_{o}) \\; \\mathbf{a}^{\\Lambda}_{L,M}\\mathbf{(r)}},\n\\end{equation}\nwhere $L$ and $M$ are the eigenvalues of the TAM and the projection of TAM operators, respectively. The $W^{\\mathrm{cir}}_{m_{l},\\lambda,p}(L,M,\\Lambda;b,w_{o})$ is the complex weight factor of the expansion which depends on the radial distance from the beam axis $b$, the beam waist $w_{o}$, projection of orbital angular momentum $m_{l}$, the radial index $p$ and the helicity $\\lambda$. The multipole expansion expresses the vector potential of the LG beam as a linear combination of electric ($\\Lambda = 1$) and magnetic ($\\Lambda = 0$) multipole components $\\mathbf{a}^{\\Lambda}_{L,M}\\mathbf{(r)}$. Mathematically, $\\mathbf{a}^{\\Lambda}_{L,M}\\mathbf{(r)}$ are expressed in terms of vector spherical harmonics of rank $L$ \\cite{reftovectorspherical,johnson2007atomic}. For circularly polarized LG beams, the complex weight factors is given by\n\\begin{widetext}\n\\begin{align}\n W^{\\mathrm{cir}}_{m_{l},\\lambda,p}(L,M,\\Lambda;b,w_{o}) &= \\sum_{L,M,\\Lambda} \\;\\sum^{p}_{\\beta=0} \\;(i \\lambda)^{\\Lambda}\\;(-1)^{\\beta} \\; 2^{\\beta +\\frac{m_{l}}{2}} \\; \\left( p+m_{l} \\atop p-\\beta \\right) \\; \\frac{(-i)^{m_{l}}w_{o}}{2\\pi}\\; \n (i)^{L+m_{l}+\\lambda-M} \\; (2 L + 1)^{1\/2} \\\\\\nonumber \n & \\times d^{L}_{M,\\lambda}(\\theta_{k}) \\;\\int_{0}^{\\infty} k_{\\bot} dk_{\\bot}\\; e^{-\\frac{k_{\\bot}^{2}w_{o}^{2}}{4}} \\left( \\frac{k_{\\bot} w_{o}}{2} \\right)^{m_{l}} \\; L^{m_{l}}_{\\beta}\\left( \\frac{(k_{\\bot} w_{o})^{2}}{4} \\right) \\; J_{m_{l}+\\lambda-M}(k_{\\bot} b).\n\\end{align}\n\\vspace{0.28in}\nUsing the complex weight factors of circularly polarized LG beams, the complex weight factor for the radially polarized LG beam reads as \n\\begin{align}\n W^{\\mathrm{rad}}_{p}(L,M,\\Lambda;b,w_{o}) = \\frac{-i}{\\sqrt{2}}\\left[ W^{\\mathrm{cir}}_{m_{l}= 1,\\lambda = -1,p}(L,M,\\Lambda;b,w_{o}) + W^{\\mathrm{cir}}_{m_{l} = -1,\\lambda = 1,p}(L,M,\\Lambda;b,w_{o}) \\right]\n\\end{align}\n\\vspace{0.28in}\nand for the azimuthally polarized LG beam as\n\\begin{align}\n W^{\\mathrm{azim}}_{p}(L,M,\\Lambda;b,w_{o}) = \\frac{1}{\\sqrt{2}}\\left[ W^{\\mathrm{cir}}_{m_{l}= 1,\\lambda = -1,p}(L,M,\\Lambda;b,w_{o}) - W^{\\mathrm{cir}}_{m_{l} = -1,\\lambda = 1,p}(L,M,\\Lambda;b,w_{o}) \\right].\n\\end{align}\n\\end{widetext}\nWith the help of complex weight factors, we can study strength of the individual multipole components of the radiation field. Since these complex weight factors depend on the radial distance $b$ and the beam waist $w_{o}$, we can control the multipole distribution of the LG beam by carefully choosing $b$ and $w_{o}$. In particular, we analyze the distribution of electric-quadrupole component and its individual projection component in the beam cross section of the LG beam in the next section. \n\n\n\\section{Results and Discussion}\n\\subsection{Distribution of electric-quadrupole field in the beam cross section of cylindrically polarized LG beam}\n\\label{subsec:Distribution of electric-quadrupole component in the beam cross section of cylindrically polarized LG beam}\n\nIn the last section, we performed multipole expansion of the vector potential of a LG beam to obtain the complex weight factors $W_{m_{l},\\lambda,p}(L,M,\\Lambda; b,w_{o})$ for both circularly and cylindrically polarized LG beams. The strength of a multipole component $L$, in particular electric-quadrupole component (L = 2), in the beam cross section is given by~$\\sum_{M=-L}^{+L}|W_{m_{l},\\lambda,p}(L=2,M,\\Lambda=1;b,w_{o})|^{2}$ for a corresponding polarization of the LG beam. We define relative strength of the electric-quadrupole~field\n\\begin{equation}\n S_{r}(E2) = \\frac{\\sum_{M=-L}^{+L}|W^{\\mathrm{cir}}_{m_{l},\\lambda,p}(L=2,M,\\Lambda=1;b,w_{o})|^{2}}{\\sum_{M=-L}^{+L}|W^{\\mathrm{rad}}_{p}(L=2,M,\\Lambda=1;b,w_{o})|^{2}}.\\label{eq:WrE2}\n\\end{equation}\nas a ratio of strength of the electric-quadrupole field between circularly polarized and radially polarized LG beams.\n\n\\par In Fig.~\\ref{fig:ratio}, we plot relative strength of the electric-quadrupole field $S_{r}(E2)$ against the radial distance $b$ from the beam axis of a LG beam for different radial index $p$ values in the top panel and the corresponding intensity profile of LG beams in bottom panel. For the radial index $p = 0$, we observe the value of $S_{r}(E2)$ to be greater than one near the beam axis ($b \\approx 0$) and decreases rapidly to less than one as radial distance $b$ increases. The behaviour of the ratio $S_{r}(E2)$ indicates that the strength of the electric-quadrupole field is suppressed near the beam axis of cylindrically polarized LG beam. However, for large $b$ values the ratio $S_{r}(E2)$ is less than one indicating a strong electric-quadrupole field in the beam cross section of cylindrically polarized LG beam. The radial index $p$ of the LG beam modifies strength of the electric-quadrupole field in the beam cross section as described by the Fig.~\\ref{fig:ratio} for $p = 1, 2$. Similar to $p = 0$ case, the ratio $S_{r}(E2)$ is greater than one near the beam axis, indicating the suppression of an electric-quadrupole field near the beam axis for cylindrically polarized LG beam. For large $b$ values, we observe the ratio $S_{r}(E2)$ to be lesser than one near the dark region in the beam cross section of the LG beam. The vertical lines in the Fig.~\\ref{fig:ratio} is used to denote the corresponding dark region in the beam cross section of LG beam with the help of intensity profile. In contrast to $p = 0$ case, cylindrically polarized LG beams possess a dominant electric-quadrupole field with respect to circularly polarized LG beam only near the dark region in the off-axis region.\n\n\\par The electric-quadrupole component can be associated with the electric field gradient of the light beam and is responsible for driving the electric-quadrupole transition in the target atoms. Therefore, the region in the Fig.~\\ref{fig:ratio} with value of $S_{r}(E2)$ less than one describes a strong electric field gradient region in the beam cross section of cylindrically polarized LG beams. This suggests that, if we were to place an atom in such a region the electric-quadrupole transition would be more efficiently driven by a cylindrical polarization over a circularly polarized LG beam.\n\\begin{figure}\n \\centering\n \\includegraphics[width=.48\\textwidth]{Figure1.pdf}\n \\caption{The relative strength of the electric-quadrupole field $S_{r}(E2)$ is plotted against the radial distance $b$ from the beam axis of the LG beam. The top plot shows variation of $S_{r}(E2)$ for three different radial indices $p = 0, 1,2$. The plot describes variation of strength of the electric-quadrupole field of radial polarization with respect to circular polarization in the beam cross section of the LG beam with beam waist of $w_{o} = 2.4$ $\\mu m$. The bottom plot describes the intensity profile of the LG beams with radial index $p = 0, 1, 2$} \n \\label{fig:ratio}\n\\end{figure}\n\\subsection{Effects of the beam waist and radial position on the projection of TAM of LG beams}\n\\label{subsec:Effect of the beam waist and radial position on the multipole distribution of LG beams}\nIn the last section, we analyzed the strength of the electric-quadrupole field in the beam cross section of the LG beam. Now, we shall discuss the strength of the projection of multipole component $M$ in the beam cross section of the LG beam with the help of modulus squared of the complex weight factor of circular or cylindrical polarization. As seen from the complex weight factors, the multipole distribution of the LG beams varies with the beam waist $w_{o}$ and the radial distance $b$ from the beam axis. The radial dependence is carried out by Bessel function $J_{m_{l}+\\lambda-M}(k_{\\bot} b)$ present in the complex weight factors. The properties of the bessel function dictates the variation of the projection of TAM $M$ across the beam cross section. For example, along the beam axis only $M = m_{l}+\\lambda$ component is non-zero because of the asymptotic property \\cite{watson1995treatise} of the Bessel function given by\n\\begin{equation}\n J_{m_{l}+\\lambda-M}(k_{\\bot} b = 0 ) = \\delta_{m_{l}+\\lambda-M,0}.\n\\end{equation}\nAs we increase the radial distance $b$ from the beam axis, the projection of TAM $M$ can have any value between $-L$ to $+L$.\n\n\\par To understand the variation of the projection of TAM across the beam cross section, we define relative weight of projection of the electric-quadrupole field as\n\\begin{equation}\n W_{r}(M) = \\frac{|W(L=2,M,\\Lambda =1;b,w_{o})|^{2}}{\\sum_{M=-L}^{+L}|W(L=2,M,\\Lambda=1;b,w_{o})|^{2}}\n\\end{equation}\nwhere $|W(L=2,M,\\Lambda=1;b,w_{o})|^{2}$ denotes the modulus squared of the complex weight factor of a circularly or cylindrically polarized LG beam.\n The $W_{r}(M)$ is plotted against the radial distance $b$ from the beam axis for both circularly and cylindrically polarized LG beams while keeping the beam waist $w_{o}$ fixed at 2.4 $\\mu m$. These plots describe the variation in the projection of the TAM across the beam cross section of the LG beams, as shown in the Fig.~\\ref{fig:cirrad} for circular and in Fig.~\\ref{fig:cylrad} for cylindrical polarization.\n\n\\par For a fixed radial distance $b$ from the beam axis, we analyze the variation of multipole distribution with respect to the beam waist $w_{o}$. For circularly polarized LG beams, $W_{r}(M)$ varies with respect to the beam waist $w_{o}$ for radial distance $b = 0.3$~$\\mu m$. However, for cylindrically polarized LG beams, $W_{r}(M)$ significantly varies with beam waist $w_{o}$ for large radial distance $b$ from the beam axis. Similarly, we plot $W_{r}(M)$ against the beam waist $w_{o}$ of the LG beams for both circular and cylindrical polarization. The Fig.~\\ref{fig:cirwaist} describes the variation in the projection of TAM $M$ in the beam cross section of circularly polarized LG beams with respect to the beam waist $w_{o}$, for $b = 0.3$ $\\mu m$. Similarly, the plots in Fig.~\\ref{fig:cylwaist} describes the variation of the projection of TAM $M$ in the beam cross section of cylindrically polarized LG beams with respect to the beam waist $w_{o}$, for $b = 2.2$ $\\mu m$.\n\n\\par By carefully selecting the beam waist $w_{o}$ and radial distance $b$ we can control the relative strength of the projection of TAM $M$ in the beam cross section of LG beam of a given polarization.\n\n\\begin{figure}\n \\centering\n \\includegraphics[width=.48\\textwidth]{Figure2.pdf}\n \\caption{The relative weight of the projection of quadrupole component $W_{r}(M)$ is plotted against the radial distance $b$ from the beam axis for right circularly polarized LG beam. In the above plots radial index $p = 0$, $\\lambda = +1$ and the radial distance $b$ is fixed to $b = 2.2$ $\\mu m$.}\n \\label{fig:cirrad}\n\\end{figure}\n\n\n\\subsection{Interaction of LG beams with the target atom}\n\\label{subsec:Interaction of LG beams with the target atom}\nThe transition between two atomic bound states occurs if the charge distribution in atomic system matches with the multipole structure of the exciting light beam. Moreover, the transition between the magnetic sub-levels in an atomic system is characterized by the projection of the TAM of the exciting light beam. Thus, the photo-excitation of atoms by LG beams helps us to analyze the multipole distribution of the twisted light beam.\n\n \\par Mathematically, the transition between initial $|n_{i},j_{i},m_{i}\\rangle$ and final $| n_{f},j_{f},m_{f}\\rangle$ bound states in an effective one electron atom is given by the transition amplitude $M_{fi}$\n \\begin{equation}\\label{eq:transition}\n M_{fi} \\; = \\; \\langle n_{f},j_{f},m_{f}| \\bm{\\alpha}\\cdot\\mathbf{A}(\\mathbf{r}) |n_{i},j_{i},m_{i}\\rangle,\n\\end{equation}\nwhere $\\mathbf{A}(r)$ is the vector potential of either circularly polarized or cylindrically polarized LG beams, $\\bm{\\alpha}$ is the Dirac matrix and the atomic bound states are characterized by principle $n$, TAM $j$ and TAM projection $m$ quantum numbers. \n\\begin{figure}\n \\centering\n \\includegraphics[width=0.48\\textwidth]{Figure3.pdf}\n \\caption{The relative weight of the projection of quadrupole component $W_{r}(M)$ is plotted against the radial distance $b$ from the beam axis for radially polarized (top) and azimuthally polarized LG beam (bottom). In the above plots radial index $p = 0$ and the beam waist is fixed to $w_{o} = 2.4$ $\\mu m$.}\n \\label{fig:cylrad}\n\\end{figure}\n\\par We substitute the multipole expansion of vector potential of LG beams into equation (\\ref{eq:transition}) to obtain the transition amplitude\n\\begin{align}\n M_{fi}(b,w_{o}) &= \\sum_{L,M,\\Lambda} W(L,M,\\Lambda;b,w_{o}) \\\\ \\nonumber \n & \\times \\langle n_{f},j_{f},m_{f}|\\bm{\\alpha} \\cdot\\mathbf{a}^{\\Lambda}_{L,M}(\\mathbf{r}) | n_{i},j_{i},m_{i} \\rangle,\n\\end{align}\nwhere $W(L,M,\\Lambda;b,w_{o})$ is the complex weight factor of either a circularly polarized or cylindrically polarized LG beam. The rest of the transition amplitude equation is solved using the Wigner-Eckart theorem \\cite{brinkangularmomentum,rose1995elementary}, which gives the transition amplitude \n \\begin{align}\\label{eq:tamplitude}\n M_{fi}(b,w_{o}) &= \\sum_{L,M,\\Lambda} W(L,M,\\Lambda;b,w_{o}) \\langle j_{i},m_{i},L,M| j_{f},m_{f}\\rangle \\\\ \\nonumber \n &\\times \\langle n_{f},j_{f}||\\bm{\\alpha}\\cdot \\mathbf{a}^{\\Lambda}_{L}(\\mathbf{r})||n_{i},j_{i}\\rangle,\n\\end{align}\nas the product of geometrical and atomic factors. The complex weight factor $W(L,M,\\Lambda;b,w_{o})$ and the Clebsch-Gordan (CG) coefficient represents the geometrical and the reduced matrix elements describe the atomic properties which influence the atomic excitation process. It is clear from the above equation (\\ref{eq:tamplitude}) that the transition amplitude depends on the complex weight factors $W(L,M,\\Lambda;b,w_{o})$, which together with the CG coefficients and reduced matrix elements determine the amplitudes of the individual transitions.\n\\par The transition amplitude of the atomic transition between two bound states must satisfy the set of following rules known as \\textit{selection rules} given by\n\\begin{equation}\n m_{i} + M = m_{f}\n\\end{equation}\n\\begin{equation}\n |j_{f}-j_{i}| \\leq L \\leq |j_{f}+j_{i}| \n\\end{equation}\n\\begin{equation}\n \\pi_{i} \\pi_{f} = (-1)^{L+p+1}\n\\end{equation}\nhere $\\pi_{i} \\pi_{f}$ are the parity of the initial and final atomic states. Selection rules are associated with the symmetry of the atomic system such as, rotational symmetry. Rotational symmetry of the atomic system dictates the conservation of TAM.\nMathematically, the selection rules are defined by symmetry relations of CG coefficients. According to the group theory, CG coefficients are a unitary transformation between an initial and final atomic states \\cite{sakurai1995modern}. These transformations or symmetry relations do not depend on the radial distance of the atoms from the beam axis. Hence, the selection rules for photo-excitation of atoms by twisted light beams are not position dependent as inferred in the previous works \\cite{selection1,duan2019,selection3}.\n\n\\subsection{Photo-excitation of Ca$^{+}$ ion by LG beams}\nWe investigate the interaction of a paraxial LG beam with beam waist $w_{o}$ and radial index $p = 0$ with the Ca$^{+}$ ion positioned at the focus along the $z$ axis, which is taken as the quantization axis. According to the electron configuration of the Ca$^{+}$ ion, only the single electron in the outer most shell has nonzero TAM. Since the lower shells have a closed configuration, their TAM is zero and does not couple with the exciting light beam. \nHence, we use one-electron notation in our work. However, to investigate the excitation of more complex atoms by twisted light beams we can use formalism of JAC \\cite{FRITZSCHE2019} to obtain the transition amplitude. \n\n\\par We consider the electric-quadrupole transition between an initial $|4s_{1\/2},m_{i}\\rangle$ of [Ar]4s state and final $|3d_{5\/2},m_{f}\\rangle$ of [Ar]3d state in Ca$^{+}$ ion. According to the selection rules, the allowed transitions between these two bound atomic states are electric-quadrupole (E2) and magnetic octopole. However, the magnetic octopole transitions are weak to be measured in an experiment and we restrict our discussion only to (E2) transitions.\n\\par The relative partial cross section $\\sigma_{r}$ is a usefull physical parameter to understand the excitation of the atoms \\cite{peshkov}, which is given by\n\\begin{equation}\n \\sigma_{r}(m_{f}) = \\frac{ |M_{fi}(b,w_{o})|^{2}} {\\sum_{m_{f}} |M_{fi}(b,w_{o})|^{2}}.\n\\end{equation}\nWe observe that the reduced matrix elements cancel in the above ratio and the above ratio now only corresponds to the product of CG coefficients and the complex weight factors. As discussed earlier, we plot only the $W_{r}(M)$ against $w_{o}$ and $b$. It is found that the extra term in the equation of relative partial cross section in the form of CG coefficients simply scales the plots of $W_{r}(M)$ against the beam waist $w_{o}$ and radial distance $b$ from the beam axis and does not change the nature of plots. Hence, studying the variation of $W_{r}(M)$ against the beam parameters is sufficient to explain the magnetic sub-level population of the target atom.\n\\begin{figure}\n \\centering\n \\includegraphics[width=0.48\\textwidth]{Figure4.pdf}\n \\caption{The relative weight of the projection of quadrupole component $W_{r}(M)$ is plotted against the beam waist of right circularly polarized LG beam. In the above plots radial index $p = 0$, $\\lambda = +1$ and the radial distance is fixed to $b = 0.3$ $\\mu m$.}\n \\label{fig:cirwaist}\n\\end{figure}\n\\subsubsection{Circularly polarized LG beams}\nBefore investigating the atomic excitation by cylindrically polarized LG beams, we consider the circularly polarized LG beams interacting with the target Ca$^{+}$ ion. As mentioned in the last section, we consider the electric-quadrupole transition between 4s and 3d states of the target Ca$^{+}$ ion. The transitions between these two states are discussed with the help of plots and the selection rule $m_{i} + M = m_{f}$.\n\n\\par In Fig.~\\ref{fig:cirrad}, we investigate the radial dependence of the atomic transition. For the target Ca$^{+}$ ion positioned along the beam axis, right circularly polarized LG beam of beam waist 2.4 $\\mu m$ drives a transition between $|4s_{1\/2}, m_{f} = - 1\/2\\rangle$ and $|3d_{5\/2}, m_{f}=3\/2\\rangle$ magnetic sub-states. Now if the target Ca$^{+}$ ion is displaced from the beam axis by some distance $b$, we observe transition between $|4s_{1\/2}, m_{f} = - 1\/2\\rangle$ and $|3d_{5\/2}, m_{f}=3\/2\\rangle$ magnetic sub-states decreases rapidly while the transition occurs between $|4s_{1\/2}, m_{f} = - 1\/2\\rangle$ and $|3d_{5\/2}, m_{f}=1\/2\\rangle$ magnetic sub-states.\n\n\\par The beam waist $w_{o}$ of the interacting circularly polarized LG beam influences the transition between the magnetic sub-states of the target Ca$^{+}$ ion positioned very close to the beam axis, as shown in the Fig.~\\ref{fig:cirwaist}. For the target Ca$^{+}$ ion displaced from the beam axis by approximately $b = 0.3$ $\\mu m$, we observe transition between $|4s_{1\/2}, m_{f} = - 1\/2\\rangle$ and $|3d_{5\/2}, m_{f}=1\/2\\rangle$ magnetic sub-states. As the beam waist $w_{o}$ of the interacting LG beam is increased, transition occurs between $|4s_{1\/2}, m_{f} = - 1\/2\\rangle$ and $|3d_{5\/2}, m_{f}=3\/2\\rangle$. The above discussion of the transition between the magnetic sub-levels of the target Ca$^{+}$ ion by a circularly polarized LG beam is in agreement with the results obtained in \\cite{peshkov} \n\n\\subsubsection{Radially polarized LG beams}\nNow we focus our attention to our main aim of this paper, that is the excitation of target atoms by cylindrically polarized LG beams. In the beginning we consider the transitions between the $4s_{1\/2}$ state and $3d_{5\/2}$ state of the Ca$^{+}$ ion driven by radially polarized LG beams of beam waist $w_{o} = 2.4$ $\\mu m$. The target Ca$^{+}$ ion positioned along the beam axis has more probablity to undergo a transition between $|4s_{1\/2}, m_{f}=-1\/2\\rangle$ and the $|3d_{5\/2}, m_{f}= -3\/2\\rangle$ magnetic sub-state absorbing $M = -1$ component, as shown in the Fig.~\\ref{fig:cylrad}. As the radial distance $b$ between the target Ca$^{+}$ ion and the beam axis is increased, we observe transition between $|4s_{1\/2}, m_{f}=-1\/2\\rangle$ and the $|3d_{5\/2}, m_{f}= 1\/2\\rangle$ magnetic sub-state absorbing $M = +1$ component.\n\\par The influence of the varying beam waist $w_{o}$ of radially polarized LG beams on the transition between the magnetic substates is discussed in the Fig.~\\ref{fig:cylwaist}. For the target Ca$^{+}$ ion displaced from the beam axis by radial distance approximately $b = 2.2$ $\\mu m$, we observe transition between $|4s_{1\/2}, m_{f} = -1\/2\\rangle$ and $|3d_{5\/2}, m_{f}= +1\/2\\rangle$ magnetic sub-state, absorbing $M = +1$ component. But as we increase the beam width, the strength of $M = +1$ component in the beam cross section of radially polarized LG beam decreases rapidly and the strength of $M = -1$ component increases. This results in the transition between $|4s_{1\/2}, m_{f} = -1\/2\\rangle$ and $|3d_{5\/2}, m_{f}= -3\/2\\rangle$ magnetic sub-states.\n\n\\subsubsection{Azimuthally polarized LG beams}\nIn Fig.~\\ref{fig:cylrad}, we discuss the photo-excitation of the target Ca$^{+}$ ion by azimuthally polarized LG beams. For the target Ca$^{+}$ ion placed along the beam axis, we observe transition between the initial $|4s_{1\/2}, m_{f} = -1\/2\\rangle$ and final $|3d_{5\/2}, m_{f}= -1\/2\\rangle$ absorbing $M = 0$ component. As the radial distance $b$ between the target Ca$^{+}$ ion is increased, strength of $M = -1$ component increases and a transition occurs between $|4s_{1\/2}, m_{f} = -1\/2\\rangle$ and $|3d_{5\/2}, m_{f}= -3\/2\\rangle$ magnetic sub-states. For large values of radial distance $b$, we observe a transition between the initial $|4s_{1\/2}, m_{f} = -1\/2\\rangle$ and final $|3d_{5\/2}, m_{f}= 1\/2\\rangle$ magnetic sub-states due to $M = +1$ component.\n\n\\par Similar to the radial polarization case, we discuss the influence of the beam waist $w_{o}$ on the transition between the magnetic sub-states in the target Ca$^{+}$ ion in the Fig.~ \\ref{fig:cylwaist}. For the target Ca$^{+}$ ion placed at a radial distance of approximately $b = 2.2$ $\\mu m$, transition occurs between initial $|4s_{1\/2}, m_{f} = -1\/2\\rangle$ and final $|3d_{5\/2}, m_{f}= 1\/2\\rangle$ magnetic sub-states absorbing $M = +1$ component. As the beam waist $w_{o}$ of the interacting azimuthally polarized LG beam is increased, strength of $M = -1$ component increases and it results in the transition between $|4s_{1\/2}, m_{f} = -1\/2\\rangle$ and $|3d_{5\/2}, m_{f}= -3\/2\\rangle$ magnetic sub-states of the target Ca$^{+}$ ion. Further, for larger values of beam waist $w_{o}$ we observe transition between $|4s_{1\/2}, m_{f} = -1\/2\\rangle$ and $|3d_{5\/2}, m_{f}= -1\/2\\rangle$ magnetic sub-states absorbing $M = 0$ component.\n\n\\begin{figure}\n \\centering\n \\includegraphics[width=0.48\\textwidth]{Figure5.pdf}\n \\caption{The relative weight of the projection of quadrupole component is plotted against the beam waist $w_{o}$ of cylindrically polarized LG beam. The plots describe the variation of strength of the projection of TAM $M$ in the beam cross section of radially (top) and azimuthally polarized (bottom) LG beams. In the above plots, radial index $p = 0$ and the radial distance is fixed to $b = 2.2$~$\\mu m$. }\n \\label{fig:cylwaist}\n\\end{figure}\n\\section{Summary}\n\\label{sec:summary}\nWe have theoretically investigated the photo-excitation of atoms by LG beams especially for cylindrical polarization. To do so, we constructed the complex weight factor of cylindrically polarized LG beam as a linear combination of complex weight factors for circular polarization. We analyzed strength of the electric-quadrupole field across the beam cross section of cylindrically polarized LG beams. We observed strength of the electric-quadrupole field of cylindrical polarization to dominate over a circularly polarized LG beam in the dark region away from the beam axis. In addition, we observed that strength of the electric-quadrupole field in the beam cross section is sensitive to the radial index $p$ of the LG beam. The variation of the magnetic component of electric-quadrupole was analyzed as a function of beam waist $w_{o}$ and the radial distance $b$ from the beam axis. To better understand the variation of the magnetic component of electric-quadrupole field, we plotted the relative weight of the projection of electric-quadrupole $W_{r}(M)$ against the beam waist and the radial distance from the beam axis.\n\\par Furthermore, we used the multipole distribution of the cylindrically polarized LG beams to discuss the excitation of target atoms. As an example, we considered the electric-quadrupole transition ($4s\\; ^{2}S_{1\/2} \\rightarrow 3d\\; ^{2}D_{5\/2}$) in the target Ca$^{+}$ ion. We observed that the radial distance $b$ from the beam axis of the cylindrically polarized LG beam influences the magnetic transitions between the $4s_{1\/2}$ and $3d_{5\/2}$ state as in the circular polarization case. Our results explicitly shows that the beam waist of the LG beams significantly affects the transitions between the magnetic sub-states of the target atoms displaced from the beam axis. Furthermore, we explicitly showed that the magnetic sub-level population in the target atom following the excitation process by LG beam can be explained primarily using the multipole distribution of the twisted light beam.\n\n\n\\begin{acknowledgments}\nThis work has been funded by the Research School of Advanced Photon Science (RS-APS) of Helmholtz Institute Jena, Germany.\n\n\\end{acknowledgments}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nEntanglement is usually presented as one of the weirdest features of\nquantum theory that depart strongly from our common\nsense~\\cite{Schrodinger:1935ys}. Since the seminal work of Einstein,\nPodolsky, and Rosen (EPR)~\\cite{Einstein:1935yt}, countless\ndiscussions on this subject have\npopped up~\\cite{Horodecki:2009vu}.\n\nA major step in the right direction is due to Bell~\\cite{Bell:1964gc}, who\nformulated the EPR dilemma in terms of an inequality which naturally\nled to a falsifiable prediction. Actually, it is common to use\nan alternative formulation, derived by Clauser, Horne, Shimony and\nHolt (CHSH)~\\cite{Clauser:1969it}, which is better suited for realistic\nexperiments.\n\nThe main stream of research~\\cite{Brunner:2014ys,Werner:2001fj} settled\nthe main concepts of this topic in the realm of quantum\nphysics. However, in recent years a general consensus has been reached\non the fact that entanglement is not necessarily a signature of the\nquantumness of a system. Actually, as aptly remarked in\nRefE.~\\cite{Toppel:2014jt}, one should distinguish between two types of\nentanglement: between spatially separated systems (inter-system\nentanglement) and between different degrees of freedom of a single\nsystem (intra-system entanglement). Inter-system entanglement occurs\nonly in truly quantum systems and may yield to nonlocal statistical\ncorrelations. Conversely, intra-system entanglement may also appear\nin classical systems and cannot generate nonlocal\ncorrelations~\\cite{Brunner:2005gv}; for this reason, it is often\ndubbed as ``classical entanglement''. Since its introduction by\nSpreeuw~\\cite{Spreeuw:1998ho}, this notion has been employed in a\nvariety of contexts~\\cite{Ghose:2014oe}. \n \nClassical entanglement has allowed to test Bell inequalities with\nclassical wave fields. The physical significance of this violation is\nnot linked to quantum nonlocality, but rather points to the\nimpossibility of constructing such a beam using other beams with\nuncoupled degrees of freedom. However, all the experiments conducted\nthus far to observe this violation have involved only discrete\nvariables, such as spin and beam path of single\nneutrons~\\cite{Hasegawa:2003hf}, polarization and transverse modes of\na laser beam~\\cite{Souza:2007fb,Simon:2010jk,Qian:2011jc,\n Gabriel:2011bn,Eberly:2014}, different transverse modes propagating\nin multimode waveguides~\\cite{Fu:2004fk}, polarization of two\nclassical fields with different frequencies~\\cite{Lee:2002uq}, orbital\nangular momentum~\\cite{Goyal:2013la,Chowdhury:2013}, and polarization\nand spatial parity~\\cite{Kagalwala:2013br}.\n\nIn this paper, we continue the analysis of this classical entanglement\nby focusing on the simple but engaging example of vortex beams. To\nthis end, in Sec.~\\ref{sec:Schmidt} we revisit a decomposition of\nLaguerre-Gauss (LG) beams in the Hermite-Gauss (HG) basis that can be\nrightly interpreted as a Schmidt decomposition. This immediately\nsuggests that many ideas ensuing from the quantum world may be\napplicable to these beams as well. In particular, in\nSec.~\\ref{sec:CHSH} we address the inseparability of the LG modes\nusing a CHSH violation that we quantify in terms of the associated\nWigner function. As this distribution can be understood as a measure\nof the displaced parity, in Sec.~\\ref{sec:exp} we discuss an\nexperimental realization which nicely agrees with the theoretical\npredictions. Finally, our conclusions are summarized in\nSec.~\\ref{sec:concl}.\n\n\n\\section{Optical vortices and Schmidt decomposition}\n\\label{sec:Schmidt}\n\nIt is well known that the beam propagation along the $z$ direction of\na monocromatic scalar field of frequency $\\omega$; i.e.,\n$E (\\mathbf{r}, t) = \\mathcal{E} (\\mathbf{r} ) \\exp [- i ( \\omega t -\nk z )] $, is governed by the paraxial wave equation\n\\begin{equation}\n \\frac{\\partial \\mathcal{E}}{\\partial z} = -\\frac{\\lambdabar}{2}\n \\left( \\frac{\\partial^2}{\\partial x^2} +\n \\frac{\\partial^2}{\\partial y^2}\\right) \\mathcal{E} \\, ,\n \\label{freespace}\n\\end{equation}\nwith $\\lambdabar = \\lambda\/2\\pi$ and $\\lambda $ is the\nwavelength. Equation~(\\ref{freespace}) is formally identical to the\nSchr\\\"{o}dinger equation for a free particle in two dimensions, with\nthe obvious identifications $t \\mapsto z$, $\\psi \\mapsto \\mathcal{E}$,\nand $\\hbar \\mapsto \\lambdabar$.\n\nAny optical beam can be thus expressed as a superposition of\nfundamental solutions of Eq.~(\\ref{freespace}). In Cartesian\ncoordinates, a natural orthonormal set is given by the Hermite-Gauss\n(HG) modes:\n\\begin{gather}\n \\mathrm{HG}_{mn} ( x, y ) = \\sqrt{\\frac{2}{\\pi n! m! 2^{n+m}}} \\left (\n \\frac{1}{w} \\right ) H_{m} \\left ( \\frac{\\sqrt{2} x}{w} \\right )\n H_{n } \\left ( \\frac{\\sqrt{2} y}{w} \\right ) \\nonumber \\\\\n \\times \\exp \\left ( - \\frac{x^{2} + y^{2}}{w^{2}} \\right ) ,\n \\label{waveHG}\n\\end{gather}\nwhere $w$ is the beam waist, and $H_{m}$ are the Hermite\npolynomials. Note that we are restricting ourselves to the plane\n$z=0$, since we are not interested here in the evolution.\n\nFor cylindrical symmetry, it is convenient to use the set of\nLaguerre-Gauss (LG) modes, which contain optical vortices with\ntopological singularities; they read\n\\begin{gather}\n \\mathrm{LG}_{mn} ( r,\\varphi ) = \\sqrt{\\frac{2}{\\pi m! n!}} \\min (m,n)!\n (-1)^{\\min (m,n)}\n \\left ( \\frac{1}{w} \\right ) \\nonumber \\\\\n \\times \\left ( \\frac{\\sqrt{2} r}{w} \\right )^{|m-n|} \\!\\! L_{\\min\n (m,n)}^{|m - n|} \\left ( \\frac{2 r^{2}}{w^{2}} \\right ) \\exp \\left\n (- \\frac{r^{2}}{w^{2}} \\right ) \\exp [ i (m-n) \\varphi ] \\, ,\n \\label{waveLG}\n\\end{gather}\nwhere $L_{p}^{|\\ell |} (x)$ are the generalized Laguerre\npolynomials. A word of caution seems to be in order: usually, these\nmodes are presented in terms of two different indices: the azimuthal\nmode index $\\ell = m - n$, which is a topological charge giving the\nnumber of $2 \\pi$-phase cycles around the mode circumference, and\n$p = \\min (m, n)$ is the radial mode index, which is related to the\nnumber of radial nodes~\\cite{Karimi:2014yu}. However, the form\n(\\ref{waveLG}) will be advantageous in what follows.\n\nThe crucial observation is that the LG modes can be represented as\nsuperpositions of HG modes, and viceversa. This can be compactly\nwritten down as~\\cite{Beijersbergen:1993eu}\n\\begin{equation}\n \\mathrm{LG}_{mn} (\\rho,\\varphi) = \\sum_{k=0}^{m+n} B_{mn}^{k} \\,\n\\mathrm{HG}_{m+n-k,k} (x,y)\n \\label{legherm}\n\\end{equation}\nwhere the coefficients are\n\\begin{equation}\nB_{mn}^{k} = \\sqrt{\\frac{k! (m +n-k)!}{m! n! 2^{n+m}}} \\frac{(-i)^{k}}{k!}\n\\frac{d^k}{dt^k} \\left . [ (1-t)^m(1+t)^n] \\right |_{t=0} .\n \\label{def11}\n\\end{equation}\nThis looks exactly the same as a Schmidt decomposition for a bipartite\nquantum system. It is nothing but a particular way of expressing a\nvector in the tensor product of two inner product\nspaces~\\cite{Peres:1993fk}. Alternatively, it can be seen as another\nform of the singular-value decomposition~\\cite{Stewart:1993uq}, which\nidentifies the maximal correlation directly. In quantum\ninformation, the Schmidt coefficients $B_{mn}^{k}$ convey complete\ninformation of the entanglement~\\cite{Agarwal:02}. Here, we intend to \n assess entanglement in LG beams via the violation of suitably\n formulated Bell inequalities. \n\n\\section{CHSH violation for Laguerre-Gauss modes}\n\\label{sec:CHSH}\n\nThe traditional form of the CHSH inequality applies to dichotomic\ndiscrete variables. For continuous variables, the sensible\nformulation is in terms of the Wigner function,\nwhich for a classical beam reads\n\\begin{equation}\n W(\\mathbf{x},\\mathbf{p}) = \\frac{1}{\\lambdabar^2 \\pi^2}\n \\int d^2\\mathbf{x}^{\\prime} \\, \n e^{2i \\mathbf{p} \\cdot \\mathbf{x}^{\\prime}\/\\lambdabar}\n \\langle E^{\\ast}(\\mathbf{x} - \\mathbf{x}^{\\prime}) E(\\mathbf{x} +\n \\mathbf{x}^{\\prime}) \\rangle \\, ,\n \\label{wigel}\n\\end{equation}\nthe angular brackets denoting statistical average. Although originally\nintroduced to represent quantum mechanical phenomena in phase\nspace~\\cite{Wigner:1932kx}, the Wigner distribution was established in\noptics~\\cite{Walther:1968qy} to relate partial coherence with\nradiometry. Since then, a great number of applications of this\nfunction have been reported~\\cite{Bastiaans:2009sp,\n Galleani:2002hb,Dragoman:1997rw,Mecklenbrauker:1997tx,Alonso:2011pi}.\nNote that $W$ has the dimensions of an intensity and it yields a\ndescription displaying both the position and the momentum (which in\nthe paraxial approximation has the significance of a scaled angular\ncoordinate) of the intensity of the wave field: in fact, one easily\nproves that\n\\begin{eqnarray}\n &\\displaystyle\n \\int W(\\mathbf{x},\\mathbf{p}) \\, d\\mathbf{p} = \n I ( \\mathbf{x} ) \\equiv \\langle E^{\\ast} (\\mathbf{x} )\n E(\\mathbf{x} \\rangle \\, , & \\nonumber \\\\\n& & \\\\\n& \\displaystyle\n \\frac{1}{\\lambdabar^2 \\pi^2} \\int W(\\mathbf{x},\\mathbf{p}) \\,\n d\\mathbf{x} = \n I ( \\mathbf{p} ) \\equiv \n \\langle E^{\\ast} (\\mathbf{p} ) E (\\mathbf{p}) \\rangle \\, , & \n\\nonumber\n\\end{eqnarray}\nwith \n\\begin{equation}\nE ( \\mathbf{p} )= \\frac{1}{\\lambdabar^2 \\pi^2} \n\\int E ( \\mathbf{x} ) \\, \\exp( i \\mathbf{p} \\cdot\n\\mathbf{x}\/\\lambdabar) \\, d\\mathbf{x} \\, .\n\\end{equation}\n Thus, the marginals of the Wigner function are the intensity\n distributions in $\\mathbf{x}$ or $\\mathbf{p}$ space, respectively. \n\n\nThe CHSH inequality can now be stated in terms of the Wigner function \n as~\\cite{Banaszek:1999mw}\n\\begin{equation}\n B = \\frac{\\pi^2}{4} | W (\\alpha, \\beta) +\n W(\\alpha,\\beta^{\\prime}) + W ( \\alpha^{\\prime},\\beta)\n - W(\\alpha^{\\prime},\\beta^{\\prime})| < 2,\n \\label{belin12}\n\\end{equation}\nwhere $\\alpha = (x, p_{x} )\/\\sqrt{2}$ and $\\beta = (y,\np_{y})\/\\sqrt{2}$. This also follows from\nthe work of Gisin~\\cite{Gisin1991201}, who formulated a Bell inequality for the\nset of observables with the property $\\hat{O}^{2} = \\openone$: as we\nshall see, the Wigner function appears as the average\nvalue of the parity, whose square is unity. Reference~\\cite{Chowdhury:2013}\npresents a detailed study of the violations of (\\ref{belin12}).\n\nFor the state $\\mathrm{LG}_{mn}$, the normalized\nWigner function can be written as~\\cite{Simon:2000xp}\n\\begin{gather}\n W_{mn}^{\\mathrm{LG}} ( X, P_{X}; Y, P_{Y} ) = \\frac{(-1)^{m+n}} {\\pi^{2}}\n \\exp(-4 Q_0) \\nonumber \\\\\n \\times L_{m} [4 (Q_0+Q_2)] L_{n} [4(Q_0-Q_2)] \\, ,\n \\label{WF_LG_n1_n2}\n\\end{gather}\nwhere \n\\begin{equation}\nQ_{0} = \\frac{1}{4} ( X^{2} + Y^{2} + P_{X}^{2} + P_{Y}^{2} ) \\, ,\n\\qquad \nQ_{2} = \\frac{1}{2} ( X P_{Y} - Y P_{X} ) \\, ,\n\\end{equation}\nand we have rescaled the variables as $ x \\mapsto (w\/\\sqrt{2}) X$\nand $p_x \\mapsto (\\sqrt{2} \\lambdabar\/w)~P_X$ (and analogously\nfor the $y$ axis). Let us first look at the simple case\nof the mode $\\mathrm{LG}_{10}$, which reduces to\n\\begin{gather}\n W^{\\mathrm{LG}}_{10} (X, P_X; Y, P_Y) = \\frac{1}{\\pi^2} \\exp(\n -P_X^2-P_Y^2-X^2-Y^2 ) \\nonumber \\\\\n\\times [ (P_X - Y)^2 + (P_Y+X)^2 -1 ] \\, .\n\\end{gather}\n The two measurement settings on one side are chosen to be\n$\\alpha = ( X = 0, P_{X}=0 ) $ and $ \\alpha^{\\prime} = (\nX^{\\prime} = X, P_{X}^{\\prime}=0 )$, and the corresponding\nsettings on the other side are $\\beta = ( Y=0, P_{Y} = 0 ) $\nand $\\beta^{\\prime} = (Y^{\\prime}=0, P_{Y}^{\\prime} =\nP_{Y})$~\\cite{Zhang:2007xb}, for which the Bell sum is\n\\begin{gather}\n B = e^{-P_Y^2} (P_Y^2 - 1 ) + e^{-X^2} ( X^2 - 1 ) \\nonumber \\\\\n - e^{-( P_Y^2 + X^2)} [ ( P_Y + X)^2 - 1] -1 \\, .\n \\label{BI_1_3}\n\\end{gather}\nUpon maximization with respect to $X$ and $P_{Y}$, we obtain the\nmaximum Bell violation, $|B_{\\mathrm{max}}| \\simeq 2.17$, which\nhappens for the choices\n$X \\simeq 0.45,~P_{Y} \\simeq 0.45$~\\cite{Chowdhury:2013}. For\ncomparison, note that the maximum Bell violation in quantum mechanics\nthrough the Wigner function for the two-mode squeezed vacuum state\nusing similar settings is given by\n$|B_{\\mathrm{max}}^{\\mathrm{QM}} | \\simeq 2.19$~\\cite{Banaszek:1999mw}.\n\nThe Bell violation may be further optimized by a more general choice\nof settings than those used here. For example, maximizing it with\nrespect to the parameters $\\alpha = (X,P_{X})$,\n$\\alpha^{\\prime}= (X^{\\prime}, P_{X}^{\\prime})$, $\\beta= (Y, P_{Y})$,\n$\\beta^{\\prime} = (Y^{\\prime}, P_{Y}^{\\prime})$, one obtains the\nabsolute maximum Bell violation, $|B_{\\max}| = 2.24$ and occurs for\nthe choices $X \\simeq -0.07,~P_{X} \\simeq 0.05,~X^{\\prime} \\simeq\n0.4,~P_{X}^{\\prime} \\simeq-0.26,~Y \\simeq-0.05,~P_{Y} \\simeq\n-0.07,~Y^{\\prime} \\simeq 0.26,~P_{Y}^{\\prime} \\simeq 0.4$.\nThe violation also increases with higher orbital angular\nmomentum. This increase with $n$ is analogous to the enhancement of\nnonlocality in quantum mechanics for many-particle\nGreenberger-Horne-Zeilinger states~\\cite{Mermin:1990nf}.\n\n\n\\section{Experimental results}\n\\label{sec:exp}\n\n\\begin{figure}[b]\n \\centerline{\\includegraphics[width=0.95\\columnwidth]{Figure1}}\n \\caption{(Color online) Scheme of the Bell measurement. The\n abbreviations are as follows: He-Ne: laser source, FC: fiber\n coupler, SMF: single mode fiber, CO: collimation optics, SLM:\n spatial light modulator, AS: aperture stop, BS: beam splitter,\n M1-M4: mirrors, CCD: camera}\n \\label{figSetup}\n\\end{figure}\n\nWe have carried a direct measurement of the Bell sums for optical\nbeams with different amount of nonlocal correlations. To understand\nthe measurement, we recall that the Wigner function in quantum optics\nis often regarded as the average of the displaced parity\noperator~\\cite{Royer:1977qf}. At the classical level, we can consider\nthe field amplitudes $\\mathcal{E} (X, Y)$ as vectors in the Hilbert\nspace of complex-valued functions that are square integrable over a\ntransverse plane. In this space we define linear Hermitian operators\n\\begin{equation}\n \\hat{X}: \\mathcal{E} (X, Y )\n \\mapsto X \\mathcal{E} (X, Y ) \\, ,\n \\qquad\n \\hat{P}_{x}: \\mathcal{E} (X, Y )\n \\mapsto - i \\frac{\\partial}{\\partial X}\n \\mathcal{E} (X, Y ) \\, ,\n\\end{equation}\nand analogous ones for the $Y$ variable. Formally, these operators\nsatisfy the canonical commutation relations\n$ [\\hat{X}, \\hat{P}_{X} ] = [\\hat{Y}, \\hat{P}_{Y} ] = i $. Therefore,\nthe unitary parity operator is \n\\begin{equation}\n\\hat{\\Pi}_{X} \\,\n\\hat{X} \\, \\hat{\\Pi}_{X} = - \\hat{X} \\, , \n\\qquad\n\\hat{\\Pi}_{X} \\, \n\\hat{P}_{X} \\,\\hat{\\Pi}_{X} = - \\hat{P}_{X} \\, ,\n\\end{equation}\nand changes $\\mathcal{E} (X, Y)$ into $\\mathcal{E} (-X, Y)$. The\ndisplacement operators are \n\\begin{equation}\n\\hat{D} (X, P_{X}) =\n\\exp [ i (P_{X} \\hat{X} - X \\hat{P}_{X} )] \\, . \n\\end{equation}\nIndeed, with these notations we have\n\\begin{align}\n W (X, P_{x}; Y, P_{y} ) & = \\nonumber \\\\\n& \\frac{4}{\\pi^2}\n \\langle\n \\hat{D} (X, P_{x}) \\hat{\\Pi}_{X} \\hat{D}^{\\dagger}(X, P_{x})\n\\, \\hat{D} (Y, P_{y}) \\hat{\\Pi}_{Y} \\hat{D}^{\\dagger}(Y, P_{y})\n \\rangle \\, .\n\\end{align}\n\n\\begin{figure}\n \\centerline{\\includegraphics[width=0.97\\columnwidth]{Figure2}}\n \\caption{(Color online) Snapshots of the CCD camera for the state\n $\\mathrm{LG}_{10}$ at the four settings $(X, P_{X}; Y, P_{Y})$ indicated. The scans are\n normalized to the peak intensity among the measurements and the\n area of interest for the intensity integration is marked by a red\n circle.}\n \\label{figCCD}\n\\end{figure}\n\n\nParity measurement can be, in turn, realized by a common-path\ninterferometer with a Dove prism inserted into the optical\npath~\\cite{Mukamel:2003qq}. In our setup, sketched in\nFig.~\\ref{figSetup}, the prism was substituted with an equivalent\nfour-mirror Sagnac arrangement~\\cite{Smith:2005vo}. The two copies of\nthe input signal obtained after the input beam splitter are\ntransformed by the mirrors so as to make one copy spatially inverted\nwith respect to the other, prior to combining the beams together. The\nresulting interference pattern is detected by a CCD camera:\nFigure~\\ref{figCCD} shows snapshots of the camera for the state\n$\\mathrm{LG}_{10}$ at the four settings indicated. The total intensity\nwitnessing parity of the measured beam is computed by spatial\nintegration and this is proportional to the desired Wigner\ndistribution sample after normalization to the overall intensity.\n\nThe target signal beams were prepared with digital holograms created\nby a spatial light modulator (SLM), which modulated a collimated\noutput of a single mode fiber coupled to a He-Ne laser. We also\nincluded a 4$f$-system, with an aperture stop, to filter the unwanted\ndiffraction orders produced by the SLM. To allow for a better\nflexibility, all the necessary shifts in the $X$, $Y$, $P_{X}$, and\n$P_{Y}$ variables were incorporated into the SLM, so that each\nBell measurement was associated with a separate hologram.\n\n\\begin{figure}\n \\centerline{\\includegraphics[width=\\columnwidth]{Figure3}}\n \\caption{(Color online) Experimental results for three different\n optical beams: a) $\\mathrm{HG}_{10}$, b)\n $0.4 \\, \\mathrm{HG}_{10}+ i 0.6 \\, \\mathrm{HG}_{01}$, and c) $\\mathrm{LG}_{10}$. At the\n top, we plot $\\frac{\\pi^{2}}{4} W(X, P_{X}; Y, P_{Y})$ at the\n values $(X,P_{X}; Y, P_{Y})$ indicated for each one. The next\n plot shows the measured Bell sums, all reported with $75\\%$ and\n $25\\%$ quartile (orange boxes) and the minimal and maximal\n measured values (error bars). The theoretical values\n $(-1.91, -2.15, -2.17)$ are the dots and the black bar is at\n $|B|=2$, which delimites the classically entangled states. The\n theoretical amplitude (top) and phase (bottom) distributions of\n the measured beams are plotted bellow the chart.}\n \\label{figResults}\n\\end{figure}\n\nThe measured beams were coherent superpositions of Hermite-Gaussian\nbeams in the form $a \\, \\mathrm{HG}_{10}+ i b \\, \\mathrm{HG}_{01}$ with\n$\\{a=1,\\, b=0\\}$, $\\{a=0.4,\\, b=0.6\\}$ and $\\{a=0.5,\\, b=0.5\\}$,\nrespectively. The first and the third are thus a pure Hermite-Gaussian\nbeam and a pure Laguerre-Gaussian vortex beam, respectively. For all\nthe beams we used the settings\n$X \\simeq 0.0,~P_{X} \\simeq 0.0,~X^{\\prime} \\simeq\n-0.45,~P_{X}^{\\prime} \\simeq 0.0,~Y \\simeq 0.0,~P_{Y} \\simeq\n0.0,~Y^{\\prime} \\simeq 0.0,~P_{Y}^{\\prime} \\simeq -0.45$\nfor the evaluation of the Bell sums. The theoretical values of the\nBell sums for these are $ (-1.91, -2.15, -2.17)$, respectively.\n\nEach measurement was repeated many times with slightly different\nreadings, due to laser intensity instabilities and CCD noise. These\neffects manifest as measurement errors, which can be estimated from\nthe sample statistics. As the parity measurement requires to normalize\nthe total measured intensity of the interference pattern with respect\nto the input beam intensity, a separate reading of the input beam\nintensity was performed. For each optical beam, the mean value of the\nBell sum is reported. The results are summarized in\nFig.~\\ref{figResults}. The Bell correlations grow with the coupling\nbetween the basis $\\mathrm{HG}_{10}$ and $\\mathrm{HG}_{01}$ modes, with statistically\nsignificant violation of CHSH inequality by the second and third\nbeams, as theoretically predicted.\n\n We also show the measured values of the Wigner function. For\n both, $\\mathrm{HG}_{10}$ and $\\mathrm{LG}_{10}$ modes, the values of\n $\\pi^2 W (0, 0; 0, 0)$ are quite close to $-1$. For classical\n beams, ours is one of the few measurements on the negativity of the\n Wigner function, though it has to be anticipated from the\n corresponding results in quantum optics~\\cite{Schleich:2000}. We note\n that very early, March and Wolf~\\cite{Wolf:1974} had constructed\n an example of a classical source which exhibited negative Wigner\n function.\n\n Finally, we have checked the violation of CHSH inequality for the\n beam $\\mathrm{LG}_{20}$. A beam with higher topological charge is more\n sensitive to setup imperfections, hence the Bell sum variation is\n significantly larger than in the case of $\\mathrm{LG}_{10}$. On the other\n hand, as shown in Fig.~\\ref{figResults20}, the increasing of the Bell\n sum for higher orbital angular momentum is clearly demonstrated:\n the theoretical value for $\\mathrm{LG}_{20}$ is $-2.24$, which agrees pretty\n well with the experimental results.\n \\footnote{A study of the Bell violations for LG beams is also presented by\n S. Prabhakar, S. G. Reddy, A. A. Chithrabhanu, P. G. K. Samantha, and\n R. P. Singh, arxiv:1406.6239, although the authors does not employ\n parity measurements, but Fourier transform.}. \nNote that the Wigner function at the origin for the $\\mathrm{LG}_{20}$ beam\nis positive, as expected. \n\n\\begin{figure}\n \\centerline{\\includegraphics[width=\\columnwidth]{Figure4}}\n \\caption{(Color online) Experimental results for the beam\n $\\mathrm{LG}_{20}$, as presented in Fig.~3. The values of $(X,0;0, P_{Y})$\n are also indicated. The Bell sum variation is significantly larger\n than in the case of $LG_{10}$. The plots on the right bottom panel\n are the amplitude and phase of $\\mathrm{LG}_{20}$.}\n \\label{figResults20}\n\\end{figure}\n\n\n\n\n\\section{Concluding remarks}\n\\label{sec:concl}\n\nIn short, we have presented an experimental study of nonlocal\ncorrelations in classical beams with topological\nsingularities~\\cite{Chowdhury:2013}. These correlations between modes are\nmanifested through the violation of a CHSH inequality, which we have\ndetected via direct parity measurements. Such a violation is shown to increase with the value of\norbital angular momentum of the beam. As a byproduct of our\nmeasurements, we obtain negativity of the Wigner function at certain\npoints in phase space for the $\\mathrm{HG}_{10}$ and $\\mathrm{LG}_{10}$ beams. Note\nthat this has implications for similar studies with electron beams,\nfor which vortices have been reported~\\cite{Schattschneider:2010,Unguris:2011}.\n\nThough entanglement here does not bear any paradoxical meaning, such\nas ``spooky action on the distance'', it still represents a\npotential resource for classical signal processing.\nIt might be expected that future applications of quantum information\nprocessing can be tailored in terms of classical light: the research\npresented in this work explores one of those options.\n\nFurthermore, our results are relevant not only for a correct understanding\nof ``classical entanglement'', but also for bringing out different\nstatistical features of the optical beams, since it provides an\nalternative paradigm to the well developed optical coherence theory.\n\n\\begin{acknowledgments}\nWe acknowledge illuminating discussions with Gerd Leuchs, Elisabeth\nGiacobino, and Andrea Aiello. This work was supported by the Grant\nAgency of the Czech Republic (Grant 15-031945), the European Social\nFund and the State Budget of the Czech Republic POSTUP II (Grant\nCZ.1.07\/2.3.00\/30.0041), the IGA of the Palack\\'y University (Grant\nPrF-2015-002), the Spanish MINECO (Grant FIS2011-26786), and UCM-Banco\nSantander Program (Grant GR3\/14).\n\\end{acknowledgments}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}