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\\ifnum0=`{\\fi\\iffalse}\\fi\n \\else\n %\n \\fi\n \\ignorespacesafterend%\n \\vspace{\\alignbot}\\par\\noindent\n} \\makeatother\n\n\n\\IEEEoverridecommandlockouts\n\n\n\n\n\\topmargin = 0em\n\n\\begin{document}\n\n\\title{Integer-Forcing Message Recovering in Interference Channels}\n\n\\author{\\authorblockN{Seyed Mohammad Azimi-Abarghouyi, Mohsen Hejazi, Behrooz Makki, Masoumeh Nasiri-Kenari, \\emph{Senior Member, IEEE}, and Tommy Svensson, \\emph{Senior Member, IEEE}}\\\\\n \\thanks{\n S.M. Azimi-Abarghouyi and M. Nasiri-Kenari are with Electrical Engineering Department, Sharif University of Technology, Tehran, Iran. Emails:\n{\\textit{azimi$\\_$sm@ee.sharif.edu, mnasiri@sharif.edu}}. M. Hejazi is with Electrical and Computer Engineering Department, University of Kashan, Kashan, Iran. Email: {\\textit{mhejazi@ee.sharif.edu}}.\nB. Makki and T. Svensson are with Department of Signals and Systems, Chalmers University\nof Technology, Gothenburg, Sweden, Emails: {\\textit{behrooz.makki@chalmers.se,\ntommy.svensson@chalmers.se}}. This work has been supported in part by VR research link project \"Green Communication via Multi-relaying\".} }\n\n\n\\maketitle\n\n\n\\begin{abstract}\nIn this paper, we propose a scheme referred to as integer-forcing message recovering (IFMR) to enable receivers to recover their desirable messages in interference channels. Compared to the state-of-the-art integer-forcing linear receiver (IFLR), our proposed IFMR approach needs to decode considerably less number of messages. In our method, each receiver recovers independent linear integer combinations of the desirable messages each from two independent equations. We propose an efficient algorithm to sequentially find the equations and integer combinations with maximum rates. We evaluate the performance of our scheme and compare the results with the minimum mean-square error (MMSE) and zero-forcing (ZF), as well as the IFLR schemes. The results indicate that our IFMR scheme outperforms the MMSE and ZF schemes, in terms of achievable rate, considerably. Also, compared to IFLR, the IFMR scheme achieves slightly less rates in moderate signal-to-noise ratios, with significantly less implementation complexity.\n\n\\end{abstract}\n\n\\vspace{-10pt}\n\\section{Introduction}\nVarious wireless communication setups can be modeled as interference channels consisting of multiple coexisting transmitter-receiver pairs. To reduce the interference in such systems, there are mainly two categories of receiver structures [1]-[2]. The first category are maximum likelihood (ML)-based receivers achieving the highest possible rates [1]. However, the ML-based estimation may be practically infeasible, as the size of the search space grows exponentially with the codeword length, the number of antennas, and the number of transmitters [1]. The second category are linear receivers (LR) which have low complexity in filtering the received signals through a linear structure for decoding. LRs are often proposed based on the criteria of zero-forcing (ZF) and minimum mean-square error (MMSE) [1]-[4]. \n\nRecently, a novel linear receiver referred to as integer-forcing linear receiver (IFLR) has been designed to simultaneously recover the transmitted messages in point-to-point multiple-input multiple-output (MIMO) systems [5]. This idea was derived from the compute-and-forward scheme [6]. Based on noisy linear combinations of the transmitted messages, IFLR recovers independent equations of messages through a linear receiver structure. In this way, in contrast to MMSE and ZF schemes, instead of combating, IFLR exploits the interference for a higher throughput. Application of the IFLR scheme in MIMO multi-pair two-way relaying is proposed in [7]. It is shown in [8] and [9] that precoding in IFLR can achieve the full diversity and the capacity of Gaussian MIMO channels up to a gap, respectively. Also, [10] applies successive decoding in IFLR and proves its sum rate optimality.\n\nIFLR recovers all desirable and undesirable transmitted messages by decoding sufficient number of the best independent equations in terms of achievable rate. Hence, considering IFLR in interference networks, the complexity of the lattice decoding and also the best equation selection process grows considerably with the number of transmitters and data streams. The combination of IFLR and interference alignment [11], referred to as integer-forcing interference alignment (IFIA), is proposed in [12] to decode sufficient equations to recover the desirable messages. However, IFIA requires channel state information at the transmitter (CSIT). This is the motivation for our paper in which we design an efficient low-complexity receiver for interference channels with no need for CSIT. \n\nHere, we propose a linear receiver scheme, referred to as integer-forcing message recovering (IFMR), for interference networks. Benefiting from a special equation structure of IFLR, we propose a novel receiver model in which the required number of decodings is limited to twice the number of desirable messages. In our IFMR, independent integer combinations of the desirable messages are recovered in each receiver. Each integer combination, referred to as desirable combined message (DCM), is recovered by decoding two independent equations. Here, with a new formulation, the equations can be optimized such that a DCM is recovered with maximum achievable rate. Despite of its much less complexity, we prove that our sequential approach in optimizing DCMs achieves the same rate as the optimal approach when we can jointly optimize DCMs (Theorem 1). \n\nInstead of NP hard exhaustive search in optimizing the equations of IFMR, we present a practical and efficient suboptimal algorithm to maximize the achievable rate in polynomial time. The proposed algorithm iterates in three steps, one for the coefficient factors of the two equations and the others for the coefficient vectors of an undesirable combined message (UCM) and DCM. The associated problem with each step is solved in polynomial time. The convergence of the proposed algorithm is also proved (Theorem 3). Hence, our IFMR scheme provides a low-complexity scheme in recovering the desirable messages through a few decodings of near-optimal integer combinations in interference channels. \n\nOur scheme is different and much less complex compared to the IFLR scheme that uses a large number of equations for message recovery. Particularly, the complexity of IFMR does not depend on the number of transmitters and the data streams of the interfering transmitters. Also, as opposed to IFIA, our scheme requires no CSIT. \n\nWe evaluate the performance of our scheme and compare the results with the minimum mean-square error (MMSE) and zero-forcing (ZF), as well as the IFLR schemes. The results indicate that, in all signal-to-noise ratios (SNRs), our IFMR scheme outperforms the MMSE and ZF schemes, in terms of achievable rate, substantially. Also, the IFMR scheme achieves slightly less rates in moderate SNRs, compared to IFLR, with significantly less implementation complexity. In addition, our proposed algorithm provides a tight lower bound for the results obtained via the NP hard exhaustive search. For instance, consider a three-pair interference channel with single antenna at the transmitters\/receivers. Then, the achievable rate of the exhaustive search is only 1 dB better than our proposed algorithm in 1 bit\/channel use. \n\nThe remainder of this paper is organized as follows. In Section II, the system model and IFLR are briefly described. Section III presents the IFMR scheme. Numerical results are given in Section IV. Finally, Section V concludes this paper.\n\n\\textbf{Notations:} The operators ${(\\mathbf{A})^*}$, $\\text{det}(\\mathbf{A})$, $\\text{Tr}(\\mathbf{A})$, $||\\mathbf{A}||$, and $\\text{span}\\left\\{\\mathbf{A}\\right\\}$ stand for conjugate transpose, determinant, trace, frobenius norm, and the space spanned by the column vectors of matrix $\\mathbf{A}$, respectively. The $\\mathbf{Z}^{n \\times 1}$ and $\\mathbf{R}^{n \\times 1}$ are the $n$ dimensional integer field and $n$ dimensional real field, respectively. Moreover, ${\\rm{lo}}{{\\rm{g}}^ + }\\left( x \\right)$ denotes ${\\rm{max}}\\left\\{ {\\log \\left( x \\right),0} \\right\\}$. The operator $\\succcurlyeq$ refers to the generalized inequality associated with the positive semidefinite cone. Also, ${\\nabla _{{\\mathbf{a}}}}{f}$ represents the partial derivative of function $f$ with respect to vector $\\mathbf{a}$. Finally, $\\mathbf{I}$ and $\\mathbf{1}$ stand for the identity matrix and the vector with all elements equal to one, respectively.\n\n\n\\section{System Model and Integer-Forcing Linear Receiver (IFLR)} \t\n\\subsection{System Model} We consider $K$-pair interference channels where $K$ transmitters are transmitting independent data streams to $K$ receivers simultaneously, as shown in Fig 1. It is assumed that there is no coordination among the transmitters and receivers. We assume no CSIT and, as a result, we do not use beamforming. This is an acceptable assumption in simple setups with no coordinations and central processing units in which channel state information (CSI) feedback and beamforming is infeasible. Incorporating partial CSIT is left for future work. In this system, the $k$-th transmitter and receiver are equipped with $N_{{t_k}}$ and $N_{{r_k}}$ antennas, respectively. The matrix $\\mathbf{H}_{kj}$ denotes the channel matrix from transmitter $k$ to receiver $j$, with dimension ${N_{{r_j}}} \\times {N_{{t_k}}}$. The elements of $\\mathbf{H}_{kj}$ are assumed to be independent identically distributed (IID) Gaussian variables with variance $\\rho _{kj}^2$. We focus on real-valued channels. However, our scheme and results are directly applicable to complex-valued channels via a real-valued decomposition, as in [5]-[6]. Transmitter $k$ exploits a lattice encoder with power constraint $P$ to map $N_{{t_k}}$ message streams $\\mathbf{w}_k$ to a real-valued codeword matrix $\\mathbf{x}_k$ with dimension ${N_{{t_k}}} \\times n$, where $n$ is the codeword length. \n\nAccording to Fig. 1, the received signal at receiver $k$ is given by\n\\begin{eqnarray}\n{\\mathbf{Y}_k} = {\\mathbf{H}_{kk}}{\\mathbf{x}_k} +\\mathop \\sum \\limits_{j = 1,j \\ne k}^K {\\mathbf{H}_{jk}}{\\mathbf{x}_j} + {\\mathbf{n}_k},\n\\end{eqnarray}\nwhere $\\mathbf{n}_k$ is IID additive white Gaussian noise with the variance $\\sigma^2, \\forall k$. \n\\subsection{Integer-Forcing Linear Receiver (IFLR)} Since the objective of our proposed approach is to limit the complexity of the IFLR scheme [5] for interference channels, it is interesting to briefly review this scheme as follows. The readers familiar with the IFLR scheme can skip this part.\n\\begin{figure}[tb!]\n\\centering\n\n\\includegraphics[width =5in]{inferifmr.png}\n\n\\caption{$K$-pair interference channel.}\n\n\\end{figure}\n\nLet us rewrite (1) as\n\\begin{eqnarray}\n{\\mathbf{Y}_k} = {\\mathbf{\\hat H}_k}\\mathbf{X} + {\\mathbf{n}_k},\n\\end{eqnarray}\nwhere ${\\mathbf{\\hat H}_k} \\buildrel \\Delta \\over = \\left[ {{\\mathbf{H}_{1k}}, \\ldots ,{\\mathbf{H}_{Kk}}} \\right]$ and $\\mathbf{X} \\buildrel \\Delta \\over = \\left[ {\\begin{array}{*{20}{c}}\n{{\\mathbf{x}_1^*}},\n \\ldots,\n{{\\mathbf{x}_K^*}}\n\\end{array}} \\right]^*$. Since $\\mathbf{X}$ is of size $L \\buildrel \\Delta \\over = \\mathop \\sum \\limits_{k = 1}^K {N_{{t_k}}}$, the IFLR scheme recovers $L$ independent equations from $\\mathbf{Y}_k$. The $L$ independent equations with equation coefficient vectors (ECVs) $\\mathbf{a}_l^k$, $l=1, \\ldots ,L$, totally shown by matrix $\\mathbf{A}^k \\buildrel \\Delta \\over = \\left[ {\\begin{array}{*{20}{c}}\n{\\mathbf{a}_1^k},\n \\ldots, \n{\\mathbf{a}_L^k}\n\\end{array}} \\right]^*$, are then solved to recover the desirable messages of the receiver $k$. \nQuantizing $\\mathbf{Y}_k$, the answer of the equation with ECV $\\mathbf{a}_l^k$ can be recovered as [6, Eq. (68)]\n\\begin{eqnarray}\n{t_l^k} = \\mathbf{a}{_l^k}^*\\mathbf{X} = Q\\left( {\\mathbf{b}{_l^k}^*{\\mathbf{Y}_k}} \\right),\n\\end{eqnarray}\nwhere $Q(\\cdot)$ denotes lattice equation quantizer, and vector $\\mathbf{b}_l^k$, of length $N_{{r_k}}$, is the projection vector. Also, $\\mathbf{b}_l^k$ is given by [5, Eq. (28)]\n\\begin{eqnarray}\n\\mathbf{b}{_l^k}^* = \\mathbf{a}{_l^k}^* \\mathbf{\\hat H}_k^*{\\left( {\\frac{1}{{\\text{SNR}}}\\mathbf{I} + {{\\mathbf{\\hat H}}_k} \\mathbf{\\hat H}_k^*} \\right)^{ - 1}},\n\\end{eqnarray}\nwhere $\\text{SNR} = \\frac{P}{{{\\sigma ^2}}}$. Finally, the rate of the equation with ECV $\\mathbf{a}_l^k$ is obtained by [5, Eq. (30)]\n\\begin{eqnarray}\nR\\left( {{\\mathbf{a}_l^k}} \\right) = {\\log ^ + }\\left( {{{\\left( {\\mathbf{a}{_l^k}^*\\left( {\\mathbf{I} - \\mathbf{\\hat H}_k^*{{\\left( {\\frac{1}{{\\text{SNR}}}\\mathbf{I} + {{\\mathbf{\\hat H}}_k} \\mathbf{\\hat H}_k^*} \\right)}^{ - 1}}{{\\mathbf{\\hat H}}_k}} \\right){\\mathbf{a}_l^k}} \\right)}^{ - 1}}} \\right).\n\\end{eqnarray}\nHence, the optimal value of $\\mathbf{A}^k$, in terms of (5), is obtained by solving the following problem\n\\begin{eqnarray}\n\\mathbf{A}_\\text{opt}^k= \\arg \\mathop {\\min}\\limits_{\\mathbf{A}^k \\in {\\mathbf{Z}^{L \\times L}}} {\\max _{l = 1, \\ldots ,L}} \\mathbf{a}{_l^k}^*\\left( {\\mathbf{I} - \\mathbf{\\hat H}_k^*{{\\left( {\\frac{1}{{\\text{SNR}}}\\mathbf{I} + {{\\mathbf{\\hat H}}_k} \\mathbf{\\hat H}_k^*} \\right)}^{ - 1}}{{ \\mathbf{\\hat H}}_k}} \\right){\\mathbf{a}_l^k},\\nonumber\\\\\n\\text{subject to} \\hspace{+10pt}\n\\text{det}(\\mathbf{A}^k) \\ne 0. \\hspace{+200pt}\n\\end{eqnarray}\nThe problem (6) is an NP hard integer programming and its complexity grows with $L$ significantly. \n\nNote that the IFLR scheme does the lattice equation quantization (3) $L$ times, which increases the implementation complexity with $L$ significantly. Hence, the IFLR scheme leads to significantly higher complexity compared to the MMSE and ZF schemes [1]-[2], i.e., $L_c \\buildrel \\Delta \\over = L - N_{{t_k}}$ more decoding for each receiver $k$.\n\nIn Section III, we propose our IFMR scheme where, independently of $K$ and ${N_{{t_i}}},\\forall i \\ne k$, each receiver $k$ only requires lattice equation decoding twice the number of the desirable messages, i.e., $2 \\times {N_{{t_k}}}$, with a low complexity best equation selection process.\n\n\n\n\n\n\\section{Integer-Forcing Message Recovering (IFMR)}\nIn summary, our proposed IFMR scheme is based on the following procedure. From the received signals ${\\mathbf{Y}_k}$ in (1), independent DCMs are recovered. For each DCM, the observed interfered signal is integer-forced to an UCM. Then, two independent equations of the DCM and UCM are decoded by the lattice quantizer as in (3) which lead to recovering the DCM. Finally, solving the recovered DCMs results in the desirable messages. \n\nIn Subsection III.A, the structure of an equation in IFMR is proposed, and accordingly its receiver model is presented. Then, in Subsection III.B, we develop a sequential three-step algorithm to efficiently find the coefficient factors of the required equations in the first step and their associated coefficient vectors of UCMs and DCMs in the second and third steps, respectively, with maximum rates in polynomial time. Theorem 1 proves that our scheme with sequential selection of DCMs achieves the same rate as the optimal scheme jointly selecting DCMs. Theorem 2 proves that Lenstra-Lenstra-Lovasz (LLL) algorithm [13] is qualified to be used for the optimization problem of the first step, and Theorem 3 proves the convergence of the proposed algorithm. Simulation results are presented in Section IV where we compare the performance of our proposed scheme with those in the literature.\n\n\\subsection{Receiver Structure} We consider an equation in the general form $t^k=d^k x_k^{\\text{DCM}}+e^k x_k^{\\text{UCM}}$ for receiver $k$, which is an integer combination of two messages $x_k^{\\text{DCM}}$ and $x_k^{\\text{UCM}}$. Here, $x_k^{{\\rm{DCM}}} \\buildrel \\Delta \\over = {\\mathbf{a}^k}^*{\\mathbf{x}_k}$ and $x_k^{{\\rm{UCM}}} \\buildrel \\Delta \\over = \\mathop \\sum \\limits_{j = 1,j \\ne k}^K \\mathbf{c}{{_j^k}^*}{\\mathbf{x}_j}$ are referred to as DCM and UCM, respectively. In other words, according to the IFLR receiver structure, $t^k$ has ECV equal to $\\left[ {\\begin{array}{*{20}{c}}\n{{d^k}{\\mathbf{a}^k}}\\\\\n{{e^k}{\\mathbf{c}^k}}\n\\end{array}} \\right]$, where ${\\mathbf{c}^k} \\buildrel \\Delta \\over = {\\left[ {\\mathbf{c}{{_1^k}^*}, \\ldots ,\\mathbf{c}{{_{k - 1}^k}^*},\\mathbf{c}{{_{k + 1}^k}^*}, \\ldots ,\\mathbf{c}{{_K^k}^*}} \\right]^*}$. $d^k$ and $e^k$ are integer coefficient factors in $\\mathbf{Z}$ space. Also, $\\mathbf{a}^k$ and\n$\\mathbf{c}_j^k, \\forall j,$ are integer coefficient vectors in ${\\mathbf{Z}^{{N_{{t_k}}} \\times 1}}$ and ${\\mathbf{Z}^{{N_{{t_j}}} \\times 1}}$, respectively. \n\nIt is straightforward to show that two equations with independent set of coefficient factors $(d_1^k,e_1^k)$ and $(d_2^k,e_2^k)$, and same $\\mathbf{a}^k$ and $\\mathbf{c}^k$ for the combined messages can obtain $x_k^{{\\rm{DCM}}} = {\\mathbf{a}^k}^*{\\mathbf{x}_k}$. According to (5) and for given coefficient vector of $x_k^{\\text{UCM}}$ and coefficient factors of the two equations, the rate of recovering $x_k^{\\text{DCM}}$ is obtained by\n\\begin{eqnarray}\n{R_{\\text{DCM}}}\\left({\\mathbf{a}^k}|{\\mathbf{c}^k},d_1^k,e_1^k,d_2^k,e_2^k\\right) = {\\rm{min}}\\left\\{ {R\\left( {\\left[ {\\begin{array}{*{20}{c}}\n{d_1^k{\\mathbf{a}^k}}\\\\\n{e_1^k{\\mathbf{c}^k}}\n\\end{array}} \\right]} \\right),R\\left( {\\left[ {\\begin{array}{*{20}{c}}\n{d_2^k{\\mathbf{a}^k}}\\\\\n{e_2^k{\\mathbf{c}^k}}\n\\end{array}} \\right]} \\right)} \\right\\},\n\\end{eqnarray}\nwith $R(\\cdot)$ given in (5). Hence, the unconditional achievable rate of $x_k^{\\text{DCM}}$ is determined by\n\\begin{eqnarray}\n{R_{\\text{DCM}}}\\left({\\mathbf{a}^k}\\right) = \\mathop {\\max }\\limits_{d_1^k,e_1^k,d_2^k,e_2^k\\in \\mathbf{Z},{\\mathbf{c}^k} \\in {\\mathbf{Z}^{{L_c} \\times 1}}} {R_{\\text{DCM}}}\\left({\\mathbf{a}^k}|{\\mathbf{c}^k},d_1^k,e_1^k,d_2^k,e_2^k\\right).\n\\end{eqnarray}\n\nDue to the size of $\\mathbf{x}_k$, it is sufficient to recover $N_{{t_k}}$ independent DCMs. An illustration of the receiver structure is given in Fig. 2. \n\\begin{figure}[tb!]\n\\centering\n\n\\includegraphics[width =5in]{recifmradv.png}\n\n\\caption{The proposed structure of receiver $k$. In each branch $i=1,...,N_{t_k}$, $\\mathbf{B}_i^k$ includes the projection vectors in (4) related to the two equations of the branch with integer coefficients $e_1^{k,i}$, $d_1^{k,i}$, $e_2^{k,i}$, $d_2^{k,i}$, $\\mathbf{a}^{k,i}$, and $\\mathbf{c}^{k,i}$. Note $\\mathbf{c}^{k,i}$, related to undesirable recovered messages, is not shown in the figure.}\n\n\\end{figure}\n\\subsection{Best Integer Coefficients Selection} From (7) and (8), it is clear that the coefficients of the optimal independent DCMs with maximum rates are jointly selected from the following optimization\n\\begin{eqnarray}\n{\\max _{d{_l^{k,m}},e{_l^{k,m}}\\in \\mathbf{Z},{\\mathbf{a}^{k,m}} \\in {\\mathbf{Z}^{{N_{{t_k}}} \\times 1}},{\\mathbf{c}^{k,m}} \\in {\\mathbf{Z}^{{L_c} \\times 1}}}}\\mathop {\\min }\\limits_{m = 1, \\ldots ,{N_{t_k}}} \\mathop {\\min }\\limits_{l = 1,2} R\\left( {\\left[ {\\begin{array}{*{20}{c}}\n{d{_l^{k,m}}\\mathbf{a}^{k,m}}\\\\\n{e{_l^{k,m}}\\mathbf{c}^{k,m}}\n\\end{array}} \\right]} \\right),\\nonumber\n\\end{eqnarray}\nsubject to\n\\begin{eqnarray}\n\\left\\{ {\\begin{array}{*{20}{c}}\n{\\text{det}\\left(\\left[ {\\begin{array}{*{20}{c}}\n{d{_1^{k,m}}}&{e{_1^{k,m}}}\\\\\n{d{_2^{k,m}}}&{e{_2^{k,m}}}\n\\end{array}} \\right]\\right) \\ne 0}, \\forall m=1,...,N_{t_k}\\\\\n{\\text{det}\\left( {\\left[ {\\mathbf{a}^{k,1}, \\ldots ,\\mathbf{a}^{k,{N_{t_k}}}} \\right]} \\right) \\ne 0}\n\\end{array}} \\right. . \n\\end{eqnarray}\nThe problem (9) is complex, because it requires searches over space $\\mathbf{Z}^{(L+2)^{N_{t_k}}\\times 1}$. For this reason, we propose a sequential selection in $N_{{t_k}}$ stages which only requires a search over space $\\mathbf{Z}^{{N_{t_k}}(L+2)\\times 1}$. Hence, the sequential scheme is of interest because it simplifies the search process, compared to (9), significantly. In the sequential selection, each stage $t$ is to recover the best DCM $x_k^{\\text{DCM},t}$ with maximum rate independently of the previously recovered messages $x_k^{\\text{DCM},j}, \\forall j < t$. To be more specific, in each stage $t$, it is required to solve\n\\begin{eqnarray}\n{\\max _{d_l^k,e_l^k \\in \\mathbf{Z},{\\mathbf{a}^k} \\in {\\mathbf{Z}^{{N_{{t_k}}} \\times 1}},{\\mathbf{c}^k} \\in {\\mathbf{Z}^{{L_c} \\times 1}}}}}\\min _{l=1,2} {R\\left( {\\left[ {\\begin{array}{*{20}{c}}\n{d_l^k{\\mathbf{a}^k}}\\\\\n{e_l^k{\\mathbf{c}^k}}\n\\end{array}} \\right]} \\right),\\nonumber\n\\end{eqnarray}\nsubject to\n\\begin{eqnarray}\n\\left\\{ {\\begin{array}{*{20}{c}}\n{\\text{det}\\left(\\left[ {\\begin{array}{*{20}{c}}\n{d_1^k}&{e_1^k}\\\\\n{d_2^k}&{e_2^k}\n\\end{array}} \\right]\\right) \\ne 0}\\\\\n{\\text{det}\\left( {\\left[ {{\\mathbf{a}^k},\\mathbf{g}_1^k, \\ldots ,\\mathbf{g}_{t - 1}^k} \\right]} \\right) \\ne 0}\n\\end{array}} \\right. , \n\\end{eqnarray}\nwhere $\\mathbf{g}_j^k$ is the integer coefficient vector associated with $x_k^{{\\text{DCM},j}}$ obtained in the stage $j R_{\\text{seq}}$. Hence, ${R_{\\text{DCM}}}\\left( {{\\mathbf{d}_{N_{t}}^k}} \\right) > {R_{\\text{DCM}}}\\left( {{\\mathbf{g}_{N_{t}}^k}} \\right)$. From (10), $\\mathbf{g}_{N_{{t}}}^k$ is obtained from two equations which have the maximum achievable rate among all set of two equations whose associated DCM coefficient vectors are linearly independent of $\\left\\{ {\\mathbf{g}_1^k}, \\ldots ,{\\mathbf{g}_{{N_t}-1}^k} \\right\\}$. This implies that every DCM coefficient vector with a rate higher than ${R_{\\text{DCM}}}\\left( \\mathbf{g}_{N_{t}}^k \\right)$ is linearly dependent to the set $\\left\\{ {{\\mathbf{g}_1^k}, \\ldots ,{\\mathbf{g}_{{N_t}-1}^k}} \\right\\}$. Thus, we conclude $\\mathbf{d}_{N_{t}}^k$ exists in the $\\text{span}\\left\\{ {\\mathbf{g}_1^k}, \\ldots ,{\\mathbf{g}_{{N_t}-1}^k} \\right\\}$. As a result, for all ${\\mathbf{d}_{{i}}^k}, \\forall i \\le N_{t}$, we have \n\\begin{eqnarray}\n\\left\\{ {{\\mathbf{d}_1^k}, \\ldots ,{\\mathbf{d}_{{N_t}}^k}} \\right\\} \\in \\text{span}\\left\\{ {{\\mathbf{g}_1^k}, \\ldots ,{\\mathbf{g}_{{N_t}-1}^k}} \\right\\},\n\\end{eqnarray}\nwhich indicates that $\\text{rank}\\left\\{ {{\\mathbf{d}_1^k}, \\ldots ,{\\mathbf{d}_{{N_t}}^k}} \\right\\} \\le {N_t}-1$. However, this contradicts the assumption of linear-independency of these equations. Hence, ${R_{\\text{opt}}} = {R_{\\text{seq}}}$.\n\n\\section{Proof of Lemma 1}\nFor every vector $\\mathbf{y} \\ne 0$ in $\\mathbf{R}^{L\\times 1}$, we can write\n\\begin{eqnarray}\n{\\bf{\\Gamma }} \\buildrel \\Delta \\over = {\\mathbf{y}^*}\\left( {{\\mathbf{I}} - \\frac{{\\mathbf{x}{\\mathbf{x}^*}}}{{{\\mathbf{x}^*}\\mathbf{x}}}} \\right)\\mathbf{y} = {\\mathbf{y}^*}\\mathbf{y} - {\\mathbf{y}^*}\\frac{{\\mathbf{x}{\\mathbf{x}^*}}}{{{\\mathbf{x}^*}\\mathbf{x}}}\\mathbf{y} = {\\mathbf{y}^*}\\mathbf{y} - \\frac{1}{{{\\mathbf{x}^*}\\mathbf{x}}}{\\mathbf{y}^*}\\mathbf{x}{\\mathbf{x}^*}\\mathbf{y}.\n\\end{eqnarray}\nThen, from the Cauchy-Schwarz inequality ${\\mathbf{y}^*}\\mathbf{x}{\\mathbf{x}^*}\\mathbf{y} \\le \\left( {{\\mathbf{y}^*}\\mathbf{y}} \\right)\\left( {{\\mathbf{x}^*}\\mathbf{x}} \\right)$, we conclude ${\\bf{\\Gamma }} \\ge 0$. Thus, ${\\mathbf{I}} - \\frac{{\\mathbf{x}{\\mathbf{x}^*}}}{{{\\mathbf{x}^*}\\mathbf{x}}}$ is semi-definite. \n\n\n\\section{Proof of Theorem 2}\nFrom the definition of $\\mathbf{U}$ in (14) and adding then subtracting a term, we can write\n\\begin{multline}\n\\mathbf{U} = \\left[ {\\begin{array}{*{20}{c}}\n{{\\mathbf{a}^k}^*{\\mathbf{a}^k}}&0\\\\\n0&{{\\mathbf{c}^k}^*{\\mathbf{c}^k}}\n\\end{array}} \\right] - \\left[ {\\begin{array}{*{20}{c}}\n{{\\mathbf{a}^k}^*\\mathbf{H}_{kk}^*}\\\\\n{{\\mathbf{c}^k}^*\\mathbf{H}_k^*}\n\\end{array}} \\right]{\\left( {\\frac{1}{{\\text{SNR}}}{\\mathbf{I}} + {\\mathbf{H}_{kk}}\\frac{{{\\mathbf{a}^k}{\\mathbf{a}^k}^*}}{{{\\mathbf{a}^k}^*{\\mathbf{a}^k}}}\\mathbf{H}_{kk}^* + {\\mathbf{H}_k}\\frac{{{\\mathbf{c}^k}{\\mathbf{c}^k}^*}}{{{\\mathbf{c}^k}^*{\\mathbf{c}^k}}}\\mathbf{H}_k^*} \\right)^{ - 1}}\\\\ \\times \\left[ {\\begin{array}{*{20}{c}}\n{{\\mathbf{H}_{kk}}{\\mathbf{a}^k}}&{{\\mathbf{H}_k}{\\mathbf{c}^k}}\n\\end{array}} \\right]+ \\left[ {\\begin{array}{*{20}{c}}\n{{\\mathbf{a}^k}^*\\mathbf{H}_{kk}^*}\\\\\n{{\\mathbf{c}^k}^*\\mathbf{H}_k^*}\n\\end{array}} \\right]{\\left( {\\frac{1}{{\\text{SNR}}}{\\mathbf{I}} + {\\mathbf{H}_{kk}}\\frac{{{\\mathbf{a}^k}{\\mathbf{a}^k}^*}}{{{\\mathbf{a}^k}^*{\\mathbf{a}^k}}}\\mathbf{H}_{kk}^* + {\\mathbf{H}_k}\\frac{{{\\mathbf{c}^k}{\\mathbf{c}^k}^*}}{{{\\mathbf{c}^k}^*{\\mathbf{c}^k}}}\\mathbf{H}_k^*} \\right)^{ - 1}} \\left[ {\\begin{array}{*{20}{c}}\n{{\\mathbf{H}_{kk}}{\\mathbf{a}^k}}&{{\\mathbf{H}_k}{\\mathbf{c}^k}}\n\\end{array}} \\right] \\\\- \\left[ {\\begin{array}{*{20}{c}}\n{{\\mathbf{a}^k}^*\\mathbf{H}_{kk}^*}\\\\\n{{\\mathbf{c}^k}^*\\mathbf{H}_k^*}\n\\end{array}} \\right]{\\left( {\\frac{1}{{\\text{SNR}}}{\\mathbf{I}} + {\\mathbf{H}_{kk}}\\mathbf{H}_{kk}^* + {\\mathbf{H}_k}\\mathbf{H}_k^*} \\right)^{ - 1}}\\left[ {\\begin{array}{*{20}{c}}\n{{\\mathbf{H}_{kk}}{\\mathbf{a}^k}}&{{\\mathbf{H}_k}{\\mathbf{c}^k}}\n\\end{array}} \\right].\n\\end{multline}\nAccording to matrix inverses identities in [20, Eqs. (159) and (165)], we can rewrite (30) as\n\\begin{multline}\n\\mathbf{U}=\\biggl( \\left[ {\\begin{array}{*{20}{c}}\n{\\frac{1}{{{\\mathbf{a}^k}^*{\\mathbf{a}^k}}}}&0\\\\\n0&{\\frac{1}{{{\\mathbf{c}^k}^*{\\mathbf{c}^k}}}}\n\\end{array}} \\right] + \\text{SNR}\\left[ {\\begin{array}{*{20}{c}}\n{\\frac{1}{{{\\mathbf{a}^k}^*{\\mathbf{a}^k}}}{\\mathbf{a}^k}^*\\mathbf{H}_{kk}^*}\\\\\n{\\frac{1}{{{\\mathbf{c}^k}^*{\\mathbf{c}^k}}}{\\mathbf{c}^k}^*\\mathbf{H}_k^*}\n\\end{array}} \\right] \\left[ {\\begin{array}{*{20}{c}}\n{\\frac{1}{{{\\mathbf{a}^k}^*{\\mathbf{a}^k}}}{\\mathbf{H}_{kk}}{\\mathbf{a}^k}}&{\\frac{1}{{{\\mathbf{c}^k}^*{\\mathbf{c}^k}}}{\\mathbf{H}_k}{\\mathbf{c}_k}}\n\\end{array}} \\right] \\biggr)^{ - 1} \\\\+ \\left[ {\\begin{array}{*{20}{c}}\n{{\\mathbf{a}^k}^*\\mathbf{H}_{kk}^*}\\\\\n{{\\mathbf{c}^k}^*\\mathbf{H}_k^*}\n\\end{array}} \\right] \\biggl\\{\\left( {\\frac{1}{{\\text{SNR}}}{\\mathbf{I}} + {\\mathbf{H}_{kk}}\\frac{{{\\mathbf{a}_k}{\\mathbf{a}_k}^*}}{{{\\mathbf{a}^k}^*{\\mathbf{a}^k}}}\\mathbf{H}_{kk}^* + {\\mathbf{H}_k}\\frac{{{\\mathbf{c}_k}{\\mathbf{c}_k}^*}}{{{\\mathbf{c}^k}^*{\\mathbf{c}^k}}}\\mathbf{H}_k^*} \\right)^{ - 1}\\\\ \\times \\left( {{\\mathbf{H}_{kk}}\\mathbf{H}_{kk}^* - {\\mathbf{H}_{kk}}\\frac{{{\\mathbf{a}_k}{\\mathbf{a}_k}^*}}{{{\\mathbf{a}^k}^*{\\mathbf{a}^k}}}\\mathbf{H}_{kk}^* + {\\mathbf{H}_k}\\mathbf{H}_k^* - {\\mathbf{H}_k}\\frac{{{\\mathbf{c}_k}{\\mathbf{c}_k}^*}}{{{\\mathbf{c}^k}^*{\\mathbf{c}^k}}}\\mathbf{H}_k^*} \\right)\n \\left( {\\frac{1}{{\\text{SNR}}}{\\mathbf{I}} + {\\mathbf{H}_{kk}}\\mathbf{H}_{kk}^* + {\\mathbf{H}_k}\\mathbf{H}_k^*} \\right)^{ - 1}\\biggr\\} \\\\ \\times \\left[ {\\begin{array}{*{20}{c}}\n{{\\mathbf{H}_{kk}}{\\mathbf{a}^k}}&{{\\mathbf{H}_k}{\\mathbf{c}_k}}\n\\end{array}} \\right].\n\\end{multline}\nIt is straightforward to show that matrices $\\mathbf{F}$, $\\mathbf{G}$, and $\\mathbf{T}$ with\n\\begin{eqnarray}\n\\mathbf{F} = \\biggl( \\left[ \\begin{array}{*{20}{c}}\n{\\frac{1}{{{\\mathbf{a}^k}^*{\\mathbf{a}^k}}}}&0\\\\\n0&{\\frac{1}{{{\\mathbf{c}^k}^*{\\mathbf{c}^k}}}}\n\\end{array} \\right] + \\text{SNR}\\left[ \\begin{array}{*{20}{c}}\n{\\frac{1}{{{\\mathbf{a}^k}^*{\\mathbf{a}^k}}}{\\mathbf{a}^k}^*\\mathbf{H}_{kk}^*}\\\\\n{\\frac{1}{{{\\mathbf{c}^k}^*{\\mathbf{c}^k}}}{\\mathbf{c}^k}^*\\mathbf{H}_k^*}\n\\end{array} \\right] \\left[ {\\begin{array}{*{20}{c}}\n{\\frac{1}{{{\\mathbf{a}^k}^*{\\mathbf{a}^k}}}{\\mathbf{H}_{kk}}{\\mathbf{a}^k}}&{\\frac{1}{{{\\mathbf{c}^k}^*{\\mathbf{c}^k}}}{\\mathbf{H}_k}{\\mathbf{c}^k}}\n\\end{array}} \\right] \\biggr)^{ - 1},\\nonumber\n\\end{eqnarray}\n\\begin{eqnarray}\n\\mathbf{G} = \\left( {\\frac{1}{{\\text{SNR}}}{\\mathbf{I}} + {\\mathbf{H}_{kk}}\\frac{{{\\mathbf{a}^k}{\\mathbf{a}^k}^*}}{{{\\mathbf{a}^k}^*{\\mathbf{a}^k}}}\\mathbf{H}_{kk}^* + {\\mathbf{H}_k}\\frac{{{\\mathbf{c}^k}{\\mathbf{c}^k}^*}}{{{\\mathbf{c}^k}^*{\\mathbf{c}^k}}}\\mathbf{H}_k^*} \\right)^{ - 1},\\nonumber\n\\end{eqnarray}\n\\begin{eqnarray}\n\\mathbf{T} = \\left( {\\frac{1}{{\\text{SNR}}}{\\mathbf{I}} + {\\mathbf{H}_{kk}}\\mathbf{H}_{kk}^* + {\\mathbf{H}_k}\\mathbf{H}_k^*} \\right)^{ - 1},\n\\end{eqnarray}\nare positive definite. \n\nAccording to Lemma 1, since ${\\mathbf{H}_{kk}}\\mathbf{H}_{kk}^* - {\\mathbf{H}_{kk}}\\frac{{{\\mathbf{a}^k}{\\mathbf{a}^k}^*}}{{{\\mathbf{a}^k}^*{\\mathbf{a}^k}}}\\mathbf{H}_{kk}^*$ and ${\\mathbf{H}_k}\\mathbf{H}_k^* - {\\mathbf{H}_k}\\frac{{{\\mathbf{c}^k}{\\mathbf{c}^k}^*}}{{{\\mathbf{c}^k}^*{\\mathbf{c}^k}}}\\mathbf{H}_k^*$ are positive semi-definite matrices, the matrix $\\mathbf{X}$ with\n\\begin{multline}\n\\mathbf{X}={\\left( {\\frac{1}{{\\text{SNR}}}{\\mathbf{I}} + {\\mathbf{H}_{kk}}\\frac{{{\\mathbf{a}^k}{\\mathbf{a}^k}^*}}{{{\\mathbf{a}^k}^*{\\mathbf{a}^k}}}\\mathbf{H}_{kk}^* + {\\mathbf{H}_k}\\frac{{{\\mathbf{c}^k}{\\mathbf{c}^k}^*}}{{{\\mathbf{c}^k}^*{\\mathbf{c}^k}}}\\mathbf{H}_k^*} \\right)^{- 1}}\\\\ \\times \\biggl( {{\\mathbf{H}_{kk}}\\mathbf{H}_{kk}^* - {\\mathbf{H}_{kk}}\\frac{{{\\mathbf{a}^k}{\\mathbf{a}^k}^*}}{{{\\mathbf{a}^k}^*{\\mathbf{a}^k}}}\\mathbf{H}_{kk}^* + {\\mathbf{H}_k}\\mathbf{H}_k^*- {\\mathbf{H}_k}\\frac{{{\\mathbf{c}^k}{\\mathbf{c}^k}^*}}{{{\\mathbf{c}^k}^*{\\mathbf{c}^k}}}\\mathbf{H}_k^*} \\biggr){\\left( {\\frac{1}{{\\text{SNR}}}{\\mathbf{I}} + {\\mathbf{H}_{kk}}\\mathbf{H}_{kk}^* + {\\mathbf{H}_k}\\mathbf{H}_k^*} \\right)^{ - 1}},\n\\end{multline}\n is also semi-definite. Hence, the overall matrix $\\mathbf{U}$, which is sum of a positive definite matrix and a semi-definite matrix, is positive definite.\n \n \n\\section{Details for the Solution of (24)}\nFor (24), we further define a function\n\\begin{eqnarray}\nV\\left( \\alpha \\right) \\buildrel \\Delta \\over = {\\rm{mi}}{{\\rm{n}}_{{\\mathbf{a}^k} \\in {\\mathbf{R}^{{N_{{t_k}}} \\times 1}}}}V\\left( {\\alpha ,{\\mathbf{a}^k}} \\right) = V\\left( {\\alpha ,{\\mathbf{a}^k}^{\\left( \\alpha \\right)}} \\right),\n\\end{eqnarray}\nwhere ${\\mathbf{a}^k}^{\\left( \\alpha \\right)}$ minimizes $V(\\alpha,{\\mathbf{a}^k})$ for given $\\alpha$. \nLet ${\\alpha ^*}$ denote the solution of ${\\rm{ma}}{{\\rm{x}}_{0 \\le \\alpha \\le 1}}V\\left( \\alpha \\right)$. There are three cases according to the relationship of ${f_1}\\left( {{\\mathbf{a}^k}^{\\left( {{\\alpha ^*}} \\right)}},{\\mathbf{c}^{{k}}} \\right)$ and ${f_2}\\left( {{\\mathbf{a}^k}^{\\left( {{\\alpha ^*}} \\right)}},{\\mathbf{c}^{{k}}} \\right)$, one of which includes the solution of (24).\n\n\\underline{Case 1}: If ${\\alpha ^*} = 0$, we have \n\\begin{eqnarray}\n{f_1}\\left( {{\\mathbf{a}^k}^{\\left( 0 \\right)},{\\mathbf{c}^{k}}} \\right) \\le {f_2}\\left( {{\\mathbf{a}^k}^{\\left( 0 \\right)},{\\mathbf{c}^{k}}} \\right).\n\\end{eqnarray}\nHence, (24) is changed to\n\\begin{eqnarray}\n\\mathop {{\\rm{min}}}\\limits_{{\\mathbf{a}^k}^{\\left( 0 \\right)} \\in {\\mathbf{R}^{{N_{{t_k}}} \\times 1}}} {f_2}\\left( {{\\mathbf{a}^k}^{\\left( 0 \\right)},{\\mathbf{c}^{k}}} \\right),\n\\end{eqnarray}\nwhich can be effectively solved by setting the derivative of ${f_2}\\left( {{\\mathbf{a}^k}^{\\left( 0 \\right)},{\\mathbf{c}^{k}}} \\right)$ with respect to ${\\mathbf{a}^k}^{\\left( 0 \\right)}$ equal to zero. Hence, according to (12), the optimal ${\\mathbf{a}^k}^{\\left( 0 \\right)}$ is given by \n\\begin{multline}\n{\\nabla _{{\\mathbf{a}^k}}}{f_2}\\left( {{\\mathbf{a}^k},{\\mathbf{c}^{k}}} \\right) = \\biggl( {d{{_2^k}^2}\\mathbf{I} - \\mathbf{H}_{kk}^*{{\\biggl( {\\frac{1}{{\\text{SNR}}}\\mathbf{I} + {\\mathbf{H}_{kk}}\\mathbf{H}_{kk}^* + {\\mathbf{H}_k}\\mathbf{H}_k^*} \\biggr)}^{ - 1}} {\\mathbf{H}_{kk}}} \\biggr){\\mathbf{a}^k}\\\\ - e_2^kd_2^k\\mathbf{H}_{kk}^*{\\left( {\\frac{1}{{\\text{SNR}}}\\mathbf{I} + {\\mathbf{H}_{kk}}\\mathbf{H}_{kk}^* + {\\mathbf{H}_k}\\mathbf{H}_k^*} \\right)^{ - 1}}{\\mathbf{H}_k}{\\mathbf{c}^{k}} = 0,\n\\end{multline}\nwhich respectively leads to\n\\begin{multline}\n{\\mathbf{a}^k}^{\\left( 0 \\right)} = {e_2^k}{d_2^k}\\biggl( d{{_2^k}^2}\\mathbf{I} - \\mathbf{H}_{kk}^*{\\left( {\\frac{1}{{\\text{SNR}}}\\mathbf{I} + {\\mathbf{H}_{kk}}\\mathbf{H}_{kk}^* + {\\mathbf{H}_k}\\mathbf{H}_k^*} \\right)}^{ - 1} {\\mathbf{H}_{kk}} \\biggr)^{ - 1} \\\\ \\times \\mathbf{H}_{kk}^*\\left( \\frac{1}{{\\text{SNR}}}\\mathbf{I} + {\\mathbf{H}_{kk}}\\mathbf{H}_{kk}^*+ {\\mathbf{H}_k}\\mathbf{H}_k^* \\right)^{ - 1}{\\mathbf{H}_k}{\\mathbf{c}^{{k}}}.\n\\end{multline}\n\\underline{Case 2}: If ${\\alpha ^*} = 1$, then we have \n\\begin{eqnarray}\n{f_1}\\left( {{\\mathbf{a}^k}^{\\left( 1 \\right)},{\\mathbf{c}^{k}}} \\right) \\ge {f_2}\\left( {{\\mathbf{a}^k}^{\\left( 1 \\right)},{\\mathbf{c}^{k}}} \\right).\n\\end{eqnarray}\nThus, similar to Case 1, we can find the ${\\mathbf{a}^k}^{\\left( 1 \\right)}$ as\n\\begin{multline}\n{\\mathbf{a}^k}^{\\left( 1 \\right)} = {e_1^k}{d_1^k}\\biggl( d{{_1^k}^2}\\mathbf{I} - \\mathbf{H}_{kk}^*{{\\left( {\\frac{1}{{\\text{SNR}}}\\mathbf{I} + {\\mathbf{H}_{kk}}\\mathbf{H}_{kk}^* + {\\mathbf{H}_k}\\mathbf{H}_k^*} \\right)}^{ - 1}}{\\mathbf{H}_{kk}} \\biggr)^{ - 1} \\\\ \\times\\mathbf{H}_{kk}^*{\\left( {\\frac{1}{{\\text{SNR}}}\\mathbf{I} + {\\mathbf{H}_{kk}}\\mathbf{H}_{kk}^* + {\\mathbf{H}_k}\\mathbf{H}_k^*} \\right)^{ - 1}}{\\mathbf{H}_k}{\\mathbf{c}^{k}}.\n\\end{multline}\n\\underline{Case 3}: If $0 < {\\alpha ^*} < 1$, then we have \n\\begin{eqnarray}\n{f_1}\\left( {{\\mathbf{a}^k}^{\\left( {{\\alpha ^*}} \\right)},{\\mathbf{c}^{k}}} \\right) = {f_2}\\left( {{\\mathbf{a}^k}^{\\left( {{\\alpha ^*}} \\right)},{\\mathbf{c}^{k}}} \\right),\n\\end{eqnarray}\nin which ${\\alpha ^*}$ can be found by the Bisection method. In this case, (24) is rephrased as\n\\begin{eqnarray}\n{\\rm{mi}}{{\\rm{n}}_{{\\mathbf{a}^k}^{\\left( {{\\alpha ^*}} \\right)} \\in {\\mathbf{R}^{{N_{{t_k}}} \\times 1}}}}V\\left( {{\\alpha ^*},{\\mathbf{a}^k}^{\\left( {{\\alpha ^*}} \\right)}} \\right) ={\\alpha ^*}{f_1}\\left( {{\\mathbf{a}^k}^{\\left( {{\\alpha ^*}} \\right)},{\\mathbf{c}^{k}}} \\right) + \\left( {1 - {\\alpha ^*}} \\right){f_2}\\left( {{\\mathbf{a}^k}^{\\left( {{\\alpha ^*}} \\right)},{\\mathbf{c}^{k}}} \\right),\n\\end{eqnarray}\nwhich can be solved by setting the derivative of $V\\left( {{\\alpha ^*},{\\mathbf{a}^k}^{\\left( {{\\alpha ^*}} \\right)}} \\right)$ with respect to ${\\mathbf{a}^k}^{\\left( {{\\alpha ^*}} \\right)}$ equal to zero. With the same arguments and using some manipulations, ${\\mathbf{a}^k}^{\\left( {{\\alpha ^*}} \\right)} $ is obtained by\n\\begin{multline}\n{\\mathbf{a}^k}^{\\left( {{\\alpha ^*}} \\right)} = u^k({\\alpha ^*}){\\left( {v^k({\\alpha ^*})\\mathbf{I} - \\mathbf{H}_{kk}^*{{\\left( {\\frac{1}{{\\text{SNR}}}\\mathbf{I} + {\\mathbf{H}_{kk}}\\mathbf{H}_{kk}^* + {\\mathbf{H}_k}\\mathbf{H}_k^*} \\right)}^{ - 1}}{\\mathbf{H}_{kk}}} \\right)^{ - 1}}\\\\ \\times \\mathbf{H}_{kk}^*{\\left( {\\frac{1}{{\\text{SNR}}}\\mathbf{I} + {\\mathbf{H}_{kk}}\\mathbf{H}_{kk}^* + {\\mathbf{H}_k}\\mathbf{H}_k^*} \\right)^{ - 1}}{\\mathbf{H}_k}{\\mathbf{c}^{k}},\n\\end{multline}\nwhere \n\\begin{eqnarray}\nu^k({\\alpha ^*}) \\buildrel \\Delta \\over = {\\alpha ^*}{e_1^k}{d_1^k} + \\left( {1 - {\\alpha ^*}} \\right){e_2^k}{d_2^k} ,\\nonumber\n\\end{eqnarray}\n\\vspace{-20pt}\n\\begin{eqnarray}\nv^k({\\alpha ^*})\\buildrel \\Delta \\over = {\\alpha ^*}d{_1^k}^2 + \\left( {1 - {\\alpha ^*}} \\right)d{_2^k}^2 .\n\\end{eqnarray}\n\n\n\n\\section{Proof of Lemma 2}\nAn equation with ECV $\\mathbf{a}_l^k$ has rate (5) equal to zero if ${\\left| {\\left| {{\\mathbf{a}_l^k}} \\right|} \\right|^2} \\geq 1+{\\rm{\\text{SNR}}}{\\lambda _{{\\rm{max}}}^2}\\left( {{{\\mathbf{\\hat H}}_k}} \\right)$ [5]. Thus, the rate (7) is zero if \n\\begin{eqnarray}\n{e_1^2\\left| {\\left| {{\\mathbf{a}^k}} \\right|} \\right|^2+d_1^2\\left| {\\left| {{\\mathbf{c}^k}} \\right|} \\right|^2} \\geq 1+{\\rm{\\text{SNR}}}{\\lambda _{{\\rm{max}}}^2}\\left( {{{\\mathbf{\\hat H}}_k}} \\right),\n\\end{eqnarray}\nor\n\\begin{eqnarray}\n {e_2^2\\left| {\\left| {{\\mathbf{a}^k}} \\right|} \\right|^2+d_2^2\\left| {\\left| {{\\mathbf{c}^k}} \\right|} \\right|^2} \\geq 1+{\\rm{\\text{SNR}}}{\\lambda _{{\\rm{max}}}^2}\\left( {{{\\mathbf{\\hat H}}_k}} \\right).\n \\end{eqnarray}\n Accordingly, we should have\\\\ \n${\\left| {\\left| {{\\mathbf{a}^k}} \\right|} \\right|^2} \\geq \\frac{1}{{e_1^2}}\\left( {1 + {\\rm{\\text{SNR}}}{\\lambda _{{\\rm{max}}}^2}\\left( {{{\\mathbf{\\hat H}}_k}} \\right) - d_1^2{\\left| {\\left| {{\\mathbf{c}^k}} \\right|} \\right|^2}} \\right)$ or $\\frac{1}{{e_2^2}}\\left( {1 + {\\rm{\\text{SNR}}}{\\lambda _{{\\rm{max}}}^2}\\left( {{{\\mathbf{\\hat H}}_k}} \\right) - d_2^2{\\left| {\\left| {{\\mathbf{c}^k}} \\right|} \\right|^2}} \\right)$, which completes the proof.\n\\section{Proof of Theorem 3}\nFor each $t$, assume $\\epsilon^j (\\mathbf{a}^{k,j},\\mathbf{c}^{k,j}) = \\mathop {{\\rm{max}}}\\limits_{l = 1,2} {f_l}^j\\left( {{\\mathbf{a}^{k,j}},{\\mathbf{c}^{k,j}}} \\right)$, where ${f_l}^j\\left( {{\\mathbf{a}^{k,j}},{\\mathbf{c}^{k,j}}} \\right)$ corresponds to the $j$-th iteration. For the iteration $j+1$ of Step I, we have $\\epsilon^{j+1} (\\mathbf{a}^{k,j},\\mathbf{c}^{k,j}) \\leq \\epsilon^j (\\mathbf{a}^{k,j},\\mathbf{c}^{k,j})$, in Step II, $\\epsilon^{j+1} (\\mathbf{a}^{k,j},\\mathbf{c}^{k,j+1} ) \\leq \\epsilon^{j+1} (\\mathbf{a}^{k,j},\\mathbf{c}^{k,j})$, and in Step III, $\\epsilon^{j+1} (\\mathbf{a}^{k,j+1},\\mathbf{c}^{k,j+1} ) \\leq \\epsilon^{j+1} (\\mathbf{a}^{k,j},\\mathbf{c}^{k,j+1})$. According to $f_l (\\mathbf{a}^k,\\mathbf{c}^k)$, the latter is guaranteed when we assume the quantization sphere has sufficiently large radius to find a suitable $\\mathbf{a}^{k,j+1}$. Even for a small quantization sphere with no candidate, we can update as $\\mathbf{a}^{k,j+1} = \\mathbf{a}^{k,j}$ which in the worst case of $\\mathbf{c}^{k,j+1} = \\mathbf{c}^{k,j}$ leads to $\\epsilon^{j+1} (\\mathbf{a}^{k,j+1},\\mathbf{c}^{k,j+1} ) = \\epsilon^{j+1} (\\mathbf{a}^{k,j},\\mathbf{c}^{k,j})$. Hence, $\\epsilon^{j+1} (\\mathbf{a}^{k,j+1},\\mathbf{c}^{k,j+1}) \\leq \\epsilon^j (\\mathbf{a}^{k,j},\\mathbf{c}^{k,j})$ at the end of iteration $j+1$. In this way, in each iteration, the function $\\epsilon = \\mathop {{\\rm{max}}}\\limits_{l = 1,2} f_l (\\mathbf{a}^k,\\mathbf{c}^k )$ either decreases or remains unchanged, and is lower bounded by zero. Thus, the proposed algorithm is convergent. \n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\nIt is accepted that the energy source that sustains the high temperature of solar corona is in the magnetic field \\citep{2006SoPh..234...41K}. Regarding the way in which this energy is converted to thermal energy, we can broadly distinguish between two scenarios that depend on the timescale of the energy release: one is continuous heating, the other is in the form of discrete, rapid pulses. The latter is consistent with the nanoflare model proposed by \\cite{1988ApJ...330..474P}. According to this model, the magnetic field tubes are displaced by the photospheric motions, and can approach and interact. When two flux tubes are almost in contact they will form a current sheet, where the field lines can reconnect. The reconnection can release a large quantity of energy in impulsive events called nanoflares. Signatures of these events are difficult to observe for several reasons, one of which is the fine structuring of the magnetic loops, that is hardly resolved with present-day instruments (e.g., \\citealt{2013ApJ...770L...1T}). \n\nOne feature to discriminate the heating release is the presence or absence of very hot plasma. In active regions, the mean coronal temperature is typically 2-3 MK. If heat pulses occur, we expect that a small amount of plasma hotter than the average (6-10 MK) will be ever-present. \n\nRecently, a number of studies, mostly based on data from the Hinode and the Solar Dynamics Observatory (SDO) missions, have shown increasing evidence for such small very hot components in active regions (\\citealt{2009ApJ...704L..58R}, \\citealt{2009ApJ...698..756R}, \\citealt{2009ApJ...697...94M}, \\citealt{2009ApJ...693L.131S}, \\citealt{2009ApJ...697.1956K}, \\citealt{2009ApJ...696..760P}, \\citealt{2010AstL...36...44S}, \\citealt{2010A&A...514A..82S}), but the issue is still debated (\\citealt{2012ApJ...754L..40T}, \\citealt{2011ApJ...734...90W}). In SDO images taken with a channel sensitive also to emission of plasma at 6 MK (94 \\AA), cores of active regions contain bright strands \\citep{2011ApJ...736L..16R}, as predicted by models of nanoflaring loops \\citep{2010ApJ...719..576G}. It remains to be proven whether this plasma is really at such high temperatures or not. Spectroscopic observations should help greatly. In this work, we analyse an active region that shows evidence for this very hot component in SDO data, but for which spectroscopic data are also available from the EUV spectrometer EIS on-board the Hinode mission. In a previous work (\\citealt{2012ApJ...750L..10T}, hereafter Paper I), emission in the CaXVII line, which forms around temperature of 6-8MK, was detected in the hot structures identified with SDO\/AIA data. In that work they built a 3 color image, to highlight the presence of a very hot component of emitting plasma inside the active region. The AIA 94~\\AA\\ band is known to be multi-thermal.\n It is sensitive to hot plasma, due to the presence of an \nFe XVIII line, formed around 6~MK, but is also sensitive to plasma at 1 MK, because of the presence of \na Fe X line and cooler Fe IX and Fe VIII (see, e.g., \\citealt{2011A&A...535A..46D}, \\citealt{2012ApJ...745..111T}, \\citealt{2011ApJ...743...23M}, \\citealt{2011ApJ...740L..52F}, \\citealt{2012A&A...537A..22O}, \\citealt{2012A&A...546A..97D}).\nRecently, also a Fe XIV line was identified in \\cite{2012A&A...546A..97D}. This line is normally stronger than the other cool components in \nactive region cores, as shown in \\cite{2013A&A...558A..73D}. It is therefore not simple to assess if the \nhot emission seen in the AIA 94~\\AA\\ band is really due to Fe XVIII. To clarify this point, \\cite{2012ApJ...750L..10T}\n compared the AIA 94~\\AA\\ image with the Ca XVII image obtained from the EIS spectrometer. They showed a strong correlation between the hot CaXVII and the emission in the 94\\AA\\ AIA band, so concluded that the hot emission seen in the AIA 94~\\AA\\ band \nis effectively due to very hot plasma (6-8MK). More direct evidence has been found from observations of another Fe XVIII line by the SUMER spectrometer on board the SOHO mission \\citep{2012ApJ...754L..40T}. \n\n\nHowever, even if the indication is rather strong, it is still not enough to establish that the plasma is actually so hot, since in theory it is possible that plasma at a lower temperature, but with very high emission measure, can give the same line intensity, as suggested by \n\\cite{2012ApJ...754L..40T}. Indeed \\cite{2013A&A...558A..73D} used simultaneous EIS and AIA observations of \nactive regions cores to show that a significant fraction of the Fe XVIII 94~\\AA\\ intensity can be due to plasma at 3~MK and not\n6~MK. In fact, \\cite{2013A&A...558A..73D} showed that often Fe XVIII 94~\\AA\\ emission is present in the cores of active regions, \nbut Ca XVII (which has a narrower formation temperature, hence is sensitive to hotter plasma) is not.\nThe only way to disentangle the various contributions to the AIA 94~\\AA\\ band is therefore to perform an emission measure modelling.\nTo this purpose, here we use the same observations from Hinode and SDO as in Paper I, that include both high-resolution spectroscopic data, over a wide spectral window, and images with high spatial resolution. \n\nAll this information provides simultaneous constraints on the plasma thermal structure along the line of sight. To further support the analysis we replicate the same analysis on the same data set but taking a region outside of the core that shows no evidence of these very hot components.We also compare two different inversion methods.\nIn Section~\\ref{sec:obs} we describe the observation and the data analysis, and in Section~\\ref{sec:discuss} the results are discussed.\n\n\\section{Observation and data analysis}\n\\label{sec:obs}\n\nWe analyse the active region (AR 11289) observed on 2011-09-13, from 10.30 to 11.30 UT. Our analysis is focused on spectroscopic data of EIS on board of Hinode \\citep{2007SoPh..243...19C}. We analyse data from a study designed by one of us (GDZ), called ATLAS\\_60, where the entire full spectral range, 178-213 \\AA\\ and 245-290 \\AA, is extracted. \nThe observations are obtained by stepping the 2$\"$ slit from solar west to east, with a 120$\"$x160$\"$ field of view. The exposure time was 60 s. The EIS data were processed using eis\\_prep, available in SolarSoft. This routine removes CCD dark current, cosmic ray strikes and takes into account hot, warm and dusty pixels. After that, radiometric calibration was applied to convert digital data (related to photon counts) into physical units (erg s$^{-1}$ cm$^{-2}$ sr$^{-1}$ \\AA). \nWe then re-aligned the fields of view by correcting for the wavelength offset of the two CCDs (by using the EIS routine eis\\_ccd\\_offset), obtained a new cropped field of view 120$\" \\times 140\"$. \n\n\n\\begin{figure*}[t!]\n\t\\centering\n\t\\includegraphics[scale=0.8]{lines.ps} \n\t\\caption{EIS, AIA and XRT images of the analysed region, i.e. 6 EIS spectral lines (Fe VIII, Fe IX, Fe XII, Fe XV, Fe XVI, Ca XVII, see Table~\\ref{tab:eislines}), 2 AIA channels (171 \\AA, 94 \\AA), and 1 XRT channel (Ti\\_poly). The position of the hot (H) and cold (C) regions analysed here are marked in all images.}\n\t\\label{fig:lines}\n\\end{figure*}\n\nIn the same temporal window we selected frames from two imagers, the X-ray telescope (XRT, \\citealt{2007SoPh..243...63G}) on board Hinode (512$\"$x512$\"$ in Ti\\_poly filter, exposure times between 0.7 and 1 s) and the Atmospheric Imaging Assembly (AIA, \\citealt{2012SoPh..275...17L}) on board SDO (full disk in 171 \\AA, 335 \\AA, 94 \\AA\\ channels, exposure times 2, 2.9, 2.9 s, respectively) . \n\nThe two sets of images were processed with the standard routines available in the Solar Soft package (xrt\\_prep and aia\\_prep). The images of the two instruments were co-aligned to each other (tr\\_get\\_disp.pro in SolarSoftware package) and to the EIS raster image in the He II 256 \\AA\\ line, to match the EIS field of view. To improve the homogeneity between the XRT or AIA images and the EIS images we have to consider that the latter are built from rastering that takes some time, while the former are \"instantaneous\". We then built-up new composite XRT and AIA images, that account for this different time spacing, as follows: each vertical strip is extracted from the images closest in time to the time when the EIS slit was at that location. XRT errors were computed taking into account the effect of the deterioration of the CCD response \\citep{2013arXiv1312.4850K}. Fig.\\ref{fig:lines} shows EIS, AIA and XRT representative images of the same field of view and with the same timing. The EIS images are obtained by integrating the spectrum in a narrow band (0.1 \\AA wide) centered at the position of each line. \n\n\nThe information available for very hot plasma from EIS data is mostly based on the Ca XVII line, the strongest EIS line formed around 6 MK \\citep{2008A&A...481L..69D}. This line is severely blended with Fe XI and O V lines, as discussed in \\cite{2007PASJ...59S.857Y}, \\cite{2009A&A...508..501D}, \\cite{2011A&A...526A...1D}, \\cite{2012ApJ...750L..10T}. As in Paper I in the case of the Ca XVII line we use the procedure developed by \\cite{2009ApJ...697.1956K} for de-blending the line from Fe XI and O V lines, and extract the Ca XVII emission.\n\nAIA and XRT images are composite images as explained above. The line images span emission from plasma in a broad temperature range $5.7 < \\log T < 6.7$. The \"cold\" ones ($\\log T < 6$) show a very bright feature consisting of \"fan\" loops in the top-left region, consistently with the image in the AIA 171 \\AA\\ channel. In the images around $\\log T = 6.4$ the morphology becomes very different: the whole region in the bottom part is quite bright and closed loops are visible, while where the bright \"cold\" features are present there is little emission.\nThat hot and cold emission is often not co-spatial has been known for a long time, and is often true even at the \nhigh spatial resolution of AIA \\citep{2013A&A...558A..73D}.\nThe Ca XVII image shows only two bright loops bifurcating southwards from a common point near the middle of the field of view. \nThese features clearly show similar morphology to the AIA 94 \\AA\\ emission, and are the same features found in Paper I. Here we also analyze XRT observations, which show that these loops are also bright in the X-rays.\nWe see also some other emission around that recalls the EIS Fe XV and Fe XVI images.\n\n\\begin{center}\n\\begin{table*}[htb!]\n\\begin{center}\n\\caption{EIS line fluxes measured after fitting each selected line with a Gaussian profile, in the two regions (cold, hot) marked in Fig.\\ref{fig:aia3col}. The ratio of the model to the observed fluxes are also reported for the DEM reconstructions with the MCMC and Del Zanna (DZ) methods (see Section~\\ref{subsec:DEMrec}).}\n\\begin{tabular}{lccccccccc} \n\\hline\\hline\n {\\bf Line}\t& Wavelength \t&\tTemperature & Cold region &Ratio &Ratio &Hot region &Ratio &Ratio \\\\\n \t& (\\AA)\t&\t ($\\log (T)$)& (erg cm$^{-2}$ sr$^{-1}$ s$^{-1}$)&DZ&MCMC&(erg cm$^{-2}$ sr$^{-1}$ s$^{-1}$)&DZ&MCMC\\\\\n \\hline\n\tFeVIII & $185.21$ & $5.7$ & $4250 \\pm 80$&1.03& 1.07 & $380 \\pm 26$&1.33&0.99\\\\\n\tMgVII &\t$278.40$ & $5.8$&$3240 \\pm 90$&0.3& 0.35& $65 \\pm 10$&0.25&0.95\\\\ \n\tFeIX & $197.862$\t& $5.9$&$380 \\pm 12$&1.00& 1.13& $27 \\pm 3$&0.97&1.09\\\\ \n\tFeX\t& $184.536$\t& $6.0$&$1186 \\pm 45$&0.89&0.81 & $255 \\pm 23$&0.94&0.58\\\\\n\tFeXI\t& $180.401$\t& $6.15$&$2270 \\pm 140$&1.07&0.83 & $ 1340 \\pm 110$&1.11&0.76\\\\\t\n\tFeXII\t& $195.119$\t& $6.20$&$1354 \\pm 20 $&1.02&1.04 & $ 1546 \\pm 21$&1.09&1.06\\\\\n\tFeXIII\t&$ 202.044$\t& $6.25$& $1001 \\pm 34 $&0.49& 0.57 & $ 1650 \\pm 40$&0.60&0.58\\\\\n\tFeXIV &$ 211.318 $& $6.3$& $1120 \\pm 100 $&0.85&1.15 & $ 3420 \\pm 170$&0.88&0.86\\\\\t \n\tFeXV\t& $284.160\t$&$ 6.35$&$3700 \\pm 120$&1.13&1.08 & $ 20070 \\pm 280$&1.17&1.07\\\\\n\tFeXVI\t& $262.984\t$&$ 6.45$&$180 \\pm 20 $&0.8&0.56& $ 2870 \\pm 80$&0.79&0.81\\\\\t \n\tSXIII\t&$ 256.685\t$& $6.45$&$399 \\pm 44 $&0.49&0.37 & $ 2380 \\pm 100$&0.78&0.65\\\\\t \n\tCaXIV & $193.866\t$&$ 6.6$&$13 \\pm 2 $&1.13&0.49 &$ 512 \\pm 13$&0.95&1.13\\\\ \n\tCaXV\t& $200.972$ &$ 6.65$&$46 \\pm 5 $&0.10& 0.05& $ 650 \\pm 20$&0.56&0.73\\\\\t \n\tCaXVII\t& $192.858$ &$ 6.70$&$0 \\pm 2 $&1.05& 0.99&$ 540 \\pm 10$&0.98&1.00\\\\ \n\\hline\n\\label{tab:eislines}\n\\end{tabular}\n\\end{center}\n\\end{table*}\n\\end{center}\n\nInside this field of view, we selected two small regions for more detailed analysis. In Paper I a special three-color coding is devised to highlight immediately hot and cool regions. In the three-color image (where each color is the intensity in a different AIA channel, green 171 \\AA, blue 335 \\AA, red 94 \\AA), we selected a strip one-pixel wide and 7-pixels long, deep inside the hot region (where the hot part of the 94 \\AA\\ emission is high), and another (equal) one inside a colder region (high 171 \\AA\\ emission, Fig.\\ref{fig:aia3col}).\n\n\\begin{figure}[t]\n\t\\centering\n\t\\includegraphics[scale= 0.6]{aia_3_colors.ps} \n\t\\caption{Image of the active region in three AIA channels: 171 \\AA\\ (green), 335 \\AA\\ (blue) and 94 \\AA\\ (red). We analyse the two small regions marked by the strips, one is hot (pink), the other is cold (green), in more detail.}\n\t\\label{fig:aia3col}\n\\end{figure}\n\n\\begin{figure*}[t]\n\t\\centering\n\t\\includegraphics[scale= 0.7]{cold_spectrum.ps} \n\t\\caption{Average Hinode\/EIS spectrum over the strip in the cold region (green in Fig.~\\ref{fig:aia3col}). The lines selected for further analysis are marked and labelled (with their temperature of maximum formation, $\\log T$). }\n\t\\label{fig:cold_spectrum}\n\\end{figure*}\n\n\\begin{figure*}[t]\n\t\\centering\n\t\\includegraphics[scale= 0.7]{hot_spectrum.ps} \n\t\\caption{As Fig.~\\ref{fig:cold_spectrum} for the strip in the Hot Region (pink in Fig.\\ref{fig:aia3col}).}\n\t\\label{fig:hot_spectrum}\n\\end{figure*}\n\nWe extracted EIS spectra in each pixel of the selected strips and, to increase the signal to noise ratio, we averaged the spectra over all the pixels in each strip. The resulting average spectra are shown in Fig.\\ref{fig:cold_spectrum} and \\ref{fig:hot_spectrum}. In these spectra the continuum is low and we see many emission lines for several elements, e.g. Fe, Ca, Mg. We selected a subset of the emission lines with the following criterion. We conceptually divided the temperature range $5.5 < \\log T < 7$ into bins $\\Delta \\log T \\sim 0.1$ and choose approximately only one line that has the maximum formation temperature in a given bin, possibly the most intense one in at least one of the two spectra. We ended up with 14 lines, listed in Table~\\ref{tab:eislines}, 9 from Fe ions, 3 from Ca, 1 from S and Mg, that provide a reasonably uniform coverage of the temperature range, as shown in Fig.~\\ref{fig:eisemis}. We use data from CHIANTI v 7.1 \\citep{2013ApJ...763...86L}. \nWe then fitted each line profile with a Gaussian (or multi-Gaussian for blended lines) and then we measured the flux by integrating the area below each Gaussian (Table~\\ref{tab:eislines}). To each flux we applied the new EIS radiometric calibration, which also includes a \ncorrection for the long-term degradation of the EIS effective area \\citep{2013A&A...555A..47D}. This leads to significantly higher (by a factor of about two) radiances of the EIS lines in the LW channel.\nOur DEM modelling is mostly constrained by strong iron lines, for which the atomic data are reliable.\nTo a first approximation, we can therefore neglect any uncertainty due to atomic data and chemical abundances.\nIt is difficult to assess the accuracy of the new EIS calibration, so we decided to \nassociate to each flux only the uncertainty due to photon statistics, and then use the results of the DEM modelling to discuss systematic errors (see Section~\\ref{sec:discuss}). We measure zero flux for the Ca XVII line in the cold region, and the value in the table is the corresponding upper limit.\n\n\\begin{figure*}[t]\n\t\\centering\n\t\\includegraphics[scale= 0.7]{emis_eis_c7.1.ps} \n\t\\caption{Emissivity of the selected lines per unit emission measure.}\n\t\\label{fig:eisemis}\n\\end{figure*}\n\n\\subsection{DEM reconstruction}\n\\label{subsec:DEMrec}\n \n We use the radiances in the selected lines (Table~\\ref{tab:eislines}) to derive the distribution of the emission measure vs temperature, DEM(T). As a first step of the analysis we use the EM loci method \\citep{1978PhDT.......298S}, by which each line intensity is divided by its emissivity. The loci of these curves represent, at each temperature, the maximum value of the emission measure at that temperature, if all the plasma were isothermal. Therefore, if the plasma were isothermal along the line of sight, all the curves would \ncross at a single point, exactly at the plasma temperature (see \\citealt{2002A&A...385..968D} for more details). There are many inversion methods to obtain the DEM.\n\nWe applied the widely-used Markov-Chain Monte Carlo method (MCMC, \\citealt{1998ApJ...503..450K}). This technique is based on a Bayesian statistical formalism to determine the most probable DEM curves that reproduce the observed line intensities by Monte Carlo simulations. A very useful feature of this technique is the possibility to obtain an estimate of the uncertainty of the DEM in each temperature bin in which the DEM is computed (see e.g., \\citealt{2011ApJ...728...30T}, \\citealt{2012ApJ...758...54T} for more detail).\n\nFor the application of the MCMC method, we assumed an electron density of $3 \\times 10^{9}$ $cm^{-3}$, \\cite{1992PhyS...46..202F} coronal abundances and CHIANTI 7.1 ionization equilibrium to compute the emissivity (per unit emission measure) of each line. The temperature range was divided into equal bins $\\Delta \\log T\\sim 0.1$. Fig.\\ref{fig:dem14} shows the result of the reconstruction in which we added the EM-loci curves. We mark the most probable solution and the cloud of solutions in each temperature bin. The broader the cloud, the less constrained is the value of the best fit curve in that bin.\n\n\\begin{figure*}[h!]\n\t\\centering\n\t\\subfigure[Cold Region]{\\includegraphics[scale= 0.6]{cold_dem.ps}}\n\t\\subfigure[Hot Region]{\\includegraphics[scale= 0.6]{hot_dem.ps}}\n\t\\caption{Results of DEM reconstruction using the MCMC technique on 14 Hinode\/EIS line fluxes (see text, and Table 1) for (a) the cold region, (b) the hot region, shown in Fig. \\ref{fig:aia3col}. The red (histogram) curve is the solution that better reproduces the observed fluxes, the histogram cloud contains the solutions. The EM-loci curves are also shown for reference, each color marks a different element.}\n\t\\label{fig:dem14}\n\\end{figure*}\n\nThere is no small region in the plot where EM-loci curves all intersect; the plasma is therefore clearly multi-thermal both in the cold and in the hot segments. The information about the very hot plasma comes mostly from the Ca lines. The very low flux of the Ca XVII line in the cold region puts a strong constraint to reduce the emission measure for $\\log T > 6.5$, while the flux is much higher in the hot region. The Fe and Ca lines show a good overall agreement while the Mg line leads to a higher emission measure in the cold region.\n\nThe MCMC method applied to the EIS data provides very different distributions in the cold and hot regions. In the cold region, the distribution is mostly flat for $\\log T < 6.4$, then it decreases rapidly. The local peak at $\\log T = 6.3$ might be significant, because of the small uncertainty. Overall the impression of a cool distribution is confirmed. In the hot region, we see a different trend: the DEM increases monotonically for $\\log T > 6$, with a slope between 2 and 3, it has a peak at $\\log T = 6.6$ and then decreases gradually at higher temperature. This shape is not new for active region core \\citep{2011ApJ...734...90W}, but our treatment of uncertainties makes the solutions better constrained. In particular, the components on the hot side of the DEM peak look quite well defined even with respect to the cooler side, where several lines have maximum formation temperatures. These hot emission measure components are at a level of $\\sim 20$\\% of the emission measure peak, therefore a significant fraction.\n\nTable~\\ref{tab:eislines} contains also the ratio of the line flux computed from the DEM reconstruction with the MCMC method to the observed one. Except for some well-known lines (Mg VII, Fe XIII) and for hot lines ($log T \\geq 6.45$) in the cold region, not relevant for our analysis, good agreement (within 20\\%) is found, comparable to that obtained in other similar analyses (e.g. \\citealt{2011ApJ...734...90W}).\n\nTo try to better constrain the high-temperature part of the DEM, first we add the information provided by the flux measured in the 94\\AA\\ AIA channel. As we briefly discussed previously, this band contains lines sensitive to emission from plasma at $\\log T \\sim 6.8$ (Fe XVIII), but also lines sensitive to lower temperatures, so the intensity in the AIA band can be written as \n\n\\begin{equation}\n\tI_{94} = EM_{cold}G_{94}(T_{cold})+EM_{hot}G_{94}(T_{hot})\n\t\\label{eq:I94comps}\n\\end{equation}\nwhere $EM_{cold}$ and $EM_{hot}$ are the emission measures of the cold and hot components, $G_{94}(T_{cold})$ and $G_{94}(T_{hot})$ are the values of the channel response functions of the cold and hot peaks respectively. \n\nWe use the technique devised by \\cite{2011ApJ...736L..16R} \n to separate the two contributions and pick up the hot one. The technique uses the flux measured in the 171 \\AA\\ channel: \n\n\\begin{equation}\n\tI_{171} = EM_{cold}G_{171}(T_{cold})\n\t\\label{eq:I171}\n\\end{equation}\n\nto constrain the cold component. Substituting Eq.\\ref{eq:I171} in Eq.\\ref{eq:I94comps} we obtain \n\n\\begin{equation}\n\tI_{94} = \\frac{I_{171}}{G_{171}(T_{cold})}G_{94}(T_{cold})+EM_{hot}G_{94}(T_{hot})\n\\end{equation}\n\nWe can then subtract the flux extrapolated from the 171 \\AA\\ channel from the 94 \\AA\\ flux. \nWhat is left is the hot part of the 94 \\AA\\ emission:\n\n\\begin{equation}\n\tI_{hot} = I_{94} - \\frac{I_{171}}{G_{171}(T_{cold})}G_{94}(T_{cold})\n\\end{equation}\n\nWe note that a similar procedure, using the 171 and 193~\\AA\\ AIA bands was devised by \n\\cite{2012ApJ...759..141W}. \\cite{2013A&A...558A..73D}\nsuggested the use of the 171 and 211~\\AA\\ AIA bands instead, the latter one being \nused to estimate the contribution of the Fe XIV line to the 94~\\AA\\ band.\nAs shown in \\cite{2013A&A...558A..73D}, this method produces \nsimilar results as the method of \\cite{2012ApJ...759..141W}.\nThe 94 \\AA\\ flux in the cold region is compatible with zero flux, so we put an upper limit as we did for the Ca XVII flux in the same region in Table~\\ref{tab:eislines}.\n\nAfter including the information from the AIA 94 \\AA\\ channel, the EM-loci AIA curves are in good agreement with the Ca XVII curves. In the hot region, the curves intersect both at $\\log T \\sim 6.6$ and at $\\log T \\sim 6.8$ but they are very similar in between. The DEM solutions derived with MCMC method are very similar to those obtained without the AIA flux, thus confirming coherent information.\n\nAdditional information about the hot components is independently available from the X-ray observation with Hinode\/X-Ray Telescope (XRT). However, we expect looser constraints from the XRT filters because they are broadband, and have a broader temperature response. We can simply plug in our analysis the flux measured in one XRT filter; in particular, we consider the Ti\\_poly filter, which has the highest sensitivity to emission from plasma at $\\log T \\sim 6.9$. The result after including both AIA and XRT fluxes is shown in Fig.\\ref{fig:dem14xrt}. We do not see qualitative difference from the DEM shown in Fig.~\\ref{fig:dem14}, as expected from the broader AIA and XRT temperature responses (see EM-loci curve). A quantitative difference is that the DEM in the hot side of the DEM peak of the hot region is reduced by a factor $\\sim 2$, i.e. to about 10\\% of the peak, while maintaining a narrow cloud distribution for $\\log T < 6.9$. \n\n\\begin{center}\n\\begin{table*}[htb!]\n\\begin{center}\n\\caption{Fluxes measured in the AIA-94 \\AA\\ channel and in the XRT-Ti\\_poly bands. \nFor the 94\\AA\\ band we subtracted the cold component, as explained in subsection \\ref{subsec:DEMrec}.}\n\\begin{tabular}{lcccc}\n\\hline\\hline\nInstrument&Cold region flux&Ratio&Hot region flux&Ratio\\\\\n& (DN s$^{-1}$ pixel$^{-1}$)&MCMC & (DN s$^{-1}$ pixel$^{-1}$)&MCMC\\\\\n\\hline\nAIA-94 \\AA&$<0.4$&0.58 &$66 \\pm 3$& 0.69 \\\\\nXRT-TI\\_poly&$53 \\pm 5$& 0.94 &$2030\t\\pm 40$& 1.28\\\\ \n\\hline\n\\label{tab:aiaxrtflx}\n\\end{tabular}\n\\end{center}\n\\end{table*}\n\\end{center}\n\n\\begin{figure*}[h]\n\t\\centering\n\t\\subfigure[Cold Region]{\\includegraphics[scale= 0.6]{cold_demaiaxrt.ps}}\n\t\\subfigure[Hot Region]{\\includegraphics[scale= 0.6]{hot_demaiaxrt.ps}}\n\t\\caption{Same as Fig.\\ref{fig:dem14} adding the contribution of the fluxes measured in the \"hot part\" (see text) of the AIA 94 \\AA\\ channel (black line) and in the XRT Ty\\_poly filter (red line).}\n\t\\label{fig:dem14xrt}\n\\end{figure*}\n\nWe further supported our analysis by comparing the best solutions of the DEM reconstruction on EIS data from the MCMC method with those from another method, devised by \\cite{1999PhDT.........8D}. \nThe method assumes that a smooth DEM distribution exists, and models it with a spline function.\nThe choice of the nodes of the spline is somewhat subjective, but the actual inversion \nis carried out following the maximum entropy method described in \\cite{1991AdSpR..11..281M}.\nWe note that the same atomic data and input parameters were used for both inversions. \nThe comparison is shown in Fig.~\\ref{fig:petdzdem14}.\n\n\t\\begin{figure*}[h!]\n\t\\centering\n\t\\subfigure[Cold Region]{\\includegraphics[scale= 0.6]{petdz_cold_dem.ps}}\n\t\\subfigure[Hot Region]{\\includegraphics[scale= 0.6]{petdz_hot_dem.ps}}\n\t\\caption{Best DEM solutions from MCMC method (histogram, red in Fig.\\ref{fig:dem14}) and from Del Zanna method (solid line $+$ symbols) for (a) the cold and (b) the hot region. The Emission Measure values with MCMC are divided by the temperature to match the output from Del Zanna method.}\n\t\\label{fig:petdzdem14}\n\t\\end{figure*}\n\nThe results with the spline method are obviously smoother, but overall there is very good agreement between the two methods for both regions, especially in the temperature ranges of interest, and in particular in the hot part of the hot region. \nGood agreement (within 20\\%) between observed and predicted intensities is found also with this method.\n\n\\section{Discussion and conclusions}\n\\label{sec:discuss}\n\nWe analysed the thermal distribution of the plasma along the line of sight in two different regions where the narrow-band imagers diagnosed very different dominant temperatures, a hotter and a cooler one. The analysis was mostly based on the spectral data from Hinode\/EIS. Our major attention was to the hottest components that may be signature of impulsive heating at work, and here our aim was to reconstruct the whole EM distribution and check the coherence of the overall scenario, including the hot component. At variance from many other similar analyses, our approach was to consider a limited number of spectral lines ($\\sim 1$ per temperature bin), and we picked the most intense lines uniformly covering the temperature range. In this way we simplified our control and interpretation of the reconstruction results, and all measured fluxes had the same weight in determining the global emission measure distribution. We also had a particular approach regarding the assessment of the uncertainties. The typical choice is to associate the same percentage error to all measured fluxes; 20\\% is the most commonly assumed value (e.g. \\citealt{2011ApJ...734...90W}). This uncertainty safely includes possible systematic unknown effects due to the instrument, atomic data and chemical abundances, but weights the same all measured values, independently of whether a line is strong or weak. Our choice is different. The measured fluxes were assigned exclusively the poissonian error, dictated only by the photon statistics. This allowed us to weight more the intense lines and to minimise the uncertainties of the solutions of the DEM reconstructions.\n\nAs a consequence, the DEM reconstruction with the MCMC technique leads to solution clouds with a narrow distribution around the best solution for many temperature bins (factor 2-3), significantly narrower than in previous works. Only a minority of temperature bins show a spread solution cloud. The constraint on the hot components is tighter. \n\nThe error estimate is important in this analysis, and we are aware that the real uncertainties are surely larger than those that we assume. The uncertainties can influence considerably the error on the DEM solution, and sometimes even the solution itself, as thoroughly discussed in \\cite{2012ApJ...745..111T}. We should also consider that the cloud spread may underestimate the real error on the DEM solution. The small uncertainties typically lead to a larger number of iterations with the MCMC method for better convergence \\citep{2012ApJ...745..111T}, but our 400 iterations is certainly an appropriate number for our cases. Tests show that DEM structures are reliably recovered on scales larger than $\\Delta \\log T = 0.2$, so we do not discuss narrower features. \n\nThe good agreement between predicted and observed line intensities confirms this, although we note that \nsome significant discrepancies are present, in particular with the Ca XV line, as also \nfound previously \\citep{2013A&A...558A..73D}. The good agreement between the two inversion methods,\nalready found in \\cite{2011A&A...535A..46D}, is confirmed. This suggests that the main source of uncertainty resides in the \nchoice of parameters, atomic data and instrument calibration.\n\nAlthough our assumptions probably underestimate the errors, the coherent support from other instruments (AIA, XRT), the agreement with the results from another reconstruction method and the coherence with the morphology seen in the images spanning the different temperature regimes makes us confident that other errors should not affect our results considerably.\n\nOverall, the analysis has confirmed the two general characteristics anticipated by the imagers. The comparison of the DEM distributions of the cold and hot regions has, on the one hand, revealed substantial thermal components for $logT < 6.3$ in the cold region, without showing prominent features. The reconstruction of the cold region also shows minor components for $logT > 6.3$. Since the images show no bright features in the hot channels and lines, we may use the values of these emission measure components as sensitivity limits of our analysis, i.e. we may not trust emission measure components below $10^{26}$ cm$^{-5}$. \n\nThe hot region shows a much more peaked thermal structure, with the positive gradient typically found in previous studies in coronal loops of active region cores \\citep{2011ApJ...734...90W}. The peak is at a rather high temperature ($\\log T = 6.6$) and beyond that the emission measure declines, but not very steeply and still showing a significant fraction of emission measure at temperature $\\log T \\leq 6.8$. Some components might be present at even higher temperature, although with higher uncertainties, up to the limit of the thermal sensitivity of our analysis ($\\log T \\sim 7$). We believe that the joint use of hot spectral lines, AIA 94 \\AA\\ channel and XRT filterbands, helps to partially remove the so-called \"blind spot\" for $\\log T > 6.8$ \\citep{2012ApJ...746L..17W}. Although the presence of the very hot component looks confirmed, still it is based on a very limited amount of information, and therefore some care should still be used. Some further feedback is provided by the comparison with the images, and between the images. The morphology of the region in the Ca XVII line only partially overlaps the morphology in immediately cooler lines (Fe XVI), while it matches well the X-ray and AIA 94 \\AA\\ morphology. \nThis confirms that in the hot region selected in the present analysis there is indeed significant \nemission at 6~MK, resulting in strong Ca XVII, so that the Fe XVIII emission in the AIA 94 \\AA\\ band is also \ndue to this component, unlike other cases in the cores of active regions where Ca XVII is not observed,\nand the Fe XVIII emission is due to a large emission measure at 3~MK \\citep{2013A&A...558A..73D}.\n\nOur analysis has remarked how spectral data are by far more constraining than the data from imagers, because the spectral lines are much more sensitive to temperature variations that the broader bands of the imagers (in agreement with the findings of \\cite{2012ApJ...758...54T}. Still it stresses that a quantum leap in the diagnostics of the hottest DEM components needs constraints from more lines sensitive to emission from high temperature plasma. These might be, for instance, easily accessible to broad-band X-ray spectrometers, to which we look forward in future space missions.\n\n\n\\acknowledgements\n{\nAP and FR acknowledge support from Italian Ministero dell'Universit\\`a e Ricerca (MIUR). GDZ acknowledges support from STFC (UK).\nPT was supported by contract SP02H1701R from Lockheed-Martin, NASA contract NNM07AB07C to SAO, and NASA grant NNX11AC20G. \\emph{CHIANTI} is a collaborative project involving the \\emph{NRL (USA)}, the \\emph{Universities of Florence (Italy)} and\n\\emph{Cambridge (UK)}, and \\emph{George Mason University (USA)}. Hinode is a Japanese mission developed and launched by ISAS\/JAXA, with NAOJ, NASA, and STFC (UK) as partners, and operated by these agencies in cooperation with ESA and NSC (Norway). SDO data were supplied courtesy of the SDO\/AIA consortium. SDO is the first mission to be launched for NASA's Living With a Star Program. \nWe thank the International Space Science Institute (ISSI) for hosting the International Team of S. Bradshaw and H. Mason: Coronal Heating \u2013 Using Observables to Settle the Question of Steady vs. Impulsive Heating.\n\n}\n\n\\bibliographystyle{aa}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\nIn the theory of quantum entanglement the familiar `distant labs' scenario motivates the restriction of allowed operations to those that can be carried out by local operations and classical communication (LOCC). Since no entangled state can be created from an unentangled one by LOCC, we regard entanglement as a resource in this context. In the case of the bipartite pure states, the picture of how this resource can be quantified and how it can be transformed (by LOCC) is quite well developed: For bipartite pure states the entanglement of distillation, $E_d$, and the entanglement cost, $E_c$, have been shown to both be equal to the entropy of entanglement (for a review see Plenio and Virmani \\cite{emeas}). Since $E_d$ and $E_c$ are equal, LOCC transformation of bipartite pure state entanglement is asymptotically reversible, and the entropy of entanglement is a unique measure of the amount of entanglement available in the many copy limit.\n\nFor mixed bipartite states, entanglement transformations are not in general asymptotically reversible, and consequently, LOCC no longer induces a unique measure of the amount of entanglement (in the asymptotic regime). For instance, there are bound entangled states (which by definition have zero entanglement of distillation) which have been shown to have non-zero entanglement cost \\cite{irrev}. This has motivated the search for non-trivial extensions of LOCC, with respect to which entanglement transformations are asymptotically reversible for all bipartite states. Recently, Plenio and Brand\\~{a}o \\cite{pb} proved that the set of operations which asymptotically cannot generate entanglement is such a set, with the regularised relative entropy of entanglement as the corresponding measure of entanglement. An open conjecture \\cite{ape} is that the set of PPT operations \\cite{rains} (defined later in this section) also renders bipartite entanglement transformations asymptotically reversible (in \\cite{ape} it is shown that it does so for the anti-symmetric Werner state).\n\nFor exact, finite transformations, the problem of determining when a given pure bipartite state can be deterministically converted into another by LOCC is completely solved by Nielsen's majorization theorem \\cite{nielsen}: The process is possible if and only if the Schmidt coefficient vector of the initial state is majorized by that of the final state.\n\nIn the present paper we investigate exact, finite transformations of pure states by PPT operations. This topic was first treated by Ishizaka and Plenio \\cite{IshizakaPlenio}), where the emphasis is on conversion of LOCC-inequivalent forms of multi-partite entanglement (e.g. GHZ and W states for three parties) by PPT operations. In the present work we are only concerned with bipartite states.\n\nHere, we introduce some conventions that will be used throughout the rest of the paper. Logarithms are always taken to base two. If the deterministic transformation of a state $\\rho$ into $\\rho'$ can be accomplished by the class of operations $OP$ then we write $\\rho \\stackrel{OP}{\\rightarrow} \\rho'$; if it cannot be then we write $\\rho \\stackrel{OP}{\\nrightarrow} \\rho'$. In discussing the transformation of bipartite pure states by any class of operations which contains LOCC, we need only consider the Schmidt coefficients of the states since states with the same Schmidt coefficients are equivalent up to local unitary transformations (which are obviously contained in LOCC). Since we are only concerned here with classes that include LOCC, we will use the state\n\n\\begin{equation}\n\t\\rho_{\\lambda} = \\sum_{i=1}^d\\sum_{j=1}^d \\sqrt{\\lambda_{i}\\lambda_{j}} \\ket{ii}\\bra{jj}\n\\end{equation}\n\n(where $\\ket{ij} = \\ket{i}_A \\otimes \\ket{j}_B$ for orthonormal bases $\\{\\ket{i}_A\\}$, $\\{\\ket{i}_B\\}$ for Alice and Bob's quotients of the bipartite Hilbert space) as a representative of all states with Schmidt coefficient vector $\\lambda$ without loss of generality. We use $\\lambda^{\\uparrow}$ ($\\lambda^{\\downarrow}$) to denote the vector obtained by putting the components of $\\lambda$ in non-decreasing (non-increasing) order.\n\nUsing this notation, Nielsen's theorem is\n\\begin{equation}\n\t\\rho_{\\lambda} \\stackrel{LOCC}{\\rightarrow} \\rho_{\\mu} \\iff \\lambda \\prec \\mu,\n\\end{equation}\nwhere the majorization relation is defined on vectors in $\\mathbb{R}^d$ whose components sum to one by\n\\begin{equation}\n\t\\lambda \\prec \\mu \\iff \\sum_{i=1}^{j}\\lambda_i^{\\downarrow} \\leq \\sum_{i=1}^{j}\\mu_i^{\\downarrow}, \\textrm{ for all } j \\in \\{1,...,d\\}.\n\\end{equation}\nWe shall use $\\Phi_K$ to denote a maximally entangled state of rank $K$ (where we assume that $K \\geq 2$).\n\nWe will use the symbol $X^{\\Gamma}$ to denote the partial transpose an operator $X$. We define the (linear) partial transpose map by\n\\begin{equation}\n\t\\ketbra{ij}{kl}^{\\Gamma} = \\ketbra{il}{kj}.\n\\end{equation}\nClearly this is basis dependent, but the eigenvalues (and hence the positivity) of the partial transpose of an operator does not depend on this basis choice. For convenience we will use the same basis that we are using as our representative Schmidt basis.\n\nThe set of PPT operations on a bipartite system is the set of completely positive trace-preserving (CPTP) maps $\\mathcal{L}$ such that the composition $\\Gamma \\circ \\mathcal{L} \\circ\\Gamma$ is also completely positive, where $\\Gamma$ denotes the partial transpose map $\\rho \\to \\rho^{\\Gamma}$. Equivalently, PPT maps are CPTP maps that \\emph{completely} preserve the PPT property of states in the same sense that completely positive maps \\emph{completely} preserve non-negativity of states: Any extension of a PPT map on a system Q onto a larger system QR where we apply the original map on Q and the identity map on R is PPT preserving (it maps PPT states to PPT states).\n\nWe make frequent use of the R\\'{e}nyi entropies: For $t \\in [0,\\infty]$ the \\emph{R\\'{e}nyi entropy at $t$} is defined by\n\n\\begin{equation}\n\t\tS_t\\left(\\lambda\\right) = \\left\\{ \\begin{array}{ll}\n\t\t\t\\frac{1}{1-t}\\log \\left( \\sum_{i=1}^{d} \\lambda_i^t \\right) &\\textrm{ for } t \\in (0,1)\\cup(1,\\infty), \\\\\n\t\t\t\\log |\\lambda| &\\textrm{ for } t = 0,\\\\\n\t\t\tH(\\lambda) &\\textrm{ for } t = 1,\\\\\n\t\t\t-\\log \\lambda^{\\downarrow}_1 &\\textrm{ for } t = \\infty,\n\t\t\\end{array} \\right.\n\\end{equation}\nwhere $|\\lambda|$ denotes the number of non-zero components of $\\lambda$ (i.e. the Schmidt rank) and $H(\\lambda) = -\\sum_i \\lambda_i\\log\\lambda_i$ is the Shannon entropy of $\\lambda$.\n\nIn the next section we discuss some necessary conditions for general bipartite pure state transformations by PPT. After that we provide part of the theory analogous to Nielsen's theory for deterministic transformations of pure bipartite states by PPT operations. In particular, we look at the case where the initial state is maximally entangled. In section \\ref{sec:frommax} we provide a necessary condition for transformations of this type, which we conjecture is also sufficient. We make use of this result in section \\ref{sec:cat} to show that the phenomenon of catalysis \\cite{catalysis} can occur for PPT operations and give a necessary and sufficient condition for when this can occur if both the initial state and catalyst state are maximally entangled. We conclude with some suggestions for future work, including a conjecture for extending this condition to arbitrary initial and catalyst states.\n\n\\section{General PPT pure state transformations}\\label{sec:general}\n\nDetermining whether a particular pure state transformation can be carried out by a PPT operation can in fact be formulated as a semidefinite programming problem \\cite{IshizakaPlenio}, the difficulty is in phrasing the relevant constraints in terms of conditions on the Schmidt coefficients of the pure states. Here we show that certain R\\'{e}nyi entropies of the Schmidt coefficient vectors correspond to operationally motivated PPT monotones (allowing us to give some necessary conditions for transformations).\n\nFor completeness let us define these operational monotones for any class of operations $X$: Let $\\rho$ be a density operator in $\\mathcal{B}(\\mathcal{H}_A\\otimes\\mathcal{H}_B)$, where $\\mathcal{B}(\\mathcal{H})$ denotes the set of hermitian operators on the Hilbert space $\\mathcal{H}$.\n\\begin{definition}\n\tThe distillable entanglement of $\\rho$ under the class of operations $X$ is defined by\n\t\\begin{equation}\n\t\tE_{d}^{{\\rm X}}( \\rho ) := \\sup_{(\\mathcal{L}_j)_{j\\in\\mathbb{N}}}\\left\\{ \\lim_{j\\to\\infty} \\frac{\\log K_j}{n_j} \\bigg| \\lim_{j\\to\\infty} \\| \\Phi_{K_j} - \\mathcal{L}_j ( \\rho^{\\otimes n_j} ) \\|_1 = 0 \\right\\},\n\t\\end{equation}\n\tand the exact distillable entanglement by\n\t\\begin{equation}\n\t\tE_{dx}^{{\\rm X}}( \\rho ) := \\lim_{j\\to\\infty} \\max_{\\mathcal{L}_j}\\left\\{ \\frac{\\log K_j}{n_j} \\bigg| \\| \\Phi_{K_j} - \\mathcal{L}_j ( \\rho^{\\otimes n_j} ) \\|_1 = 0 \\right\\},\n\t\\end{equation}\n\twhere in both cases $(\\mathcal{L}_j)_{j\\in\\mathbb{N}}$ is a sequence of completely positive trace preserving (CPTP) maps such that each map, $\\mathcal{L}_j$, takes $\\mathcal{B}\\left(\\left(\\mathcal{H}_A\\otimes\\mathcal{H}_B\\right)^{\\otimes n_j}\\right)$ to $\\mathcal{B}\\left(\\mathbb{C}^{\\otimes K_j}\\otimes\\mathbb{C}^{\\otimes K_j}\\right)$ and belongs to the class $X$.\n\\end{definition}\n\n\\begin{definition}\n\tThe entanglement cost of a state $\\rho$, we define by\n\t\\begin{equation}\n\t\tE_{c}^{{\\rm X}}( \\rho ) := \\inf_{(\\mathcal{L}_j)_{j\\in\\mathbb{N}}}\\left\\{ \\lim_{j\\to\\infty} \\frac{\\log K_j}{n_j} \\bigg| \\lim_{j\\to\\infty} \\| \\mathcal{L}_j (\\Phi_{K_j} ) - \\rho^{\\otimes n_j} \\|_1 = 0 \\right\\},\n\t\\end{equation}\n\tand the exact entanglement cost by\n\t\\begin{equation}\n\t\tE_{cx}^{{\\rm X}}( \\rho ) := \\lim_{j\\to\\infty} \\min_{\\mathcal{L}_j}\\left\\{ \\frac{\\log K_j}{n_j} \\bigg| \\| \\mathcal{L}_j (\\Phi_{K_j} ) - \\rho^{\\otimes n_j} \\|_1 = 0 \\right\\},\n\t\\end{equation}\n\twhere in both cases $(\\mathcal{L}_j)_{j\\in\\mathbb{N}}$ is a sequence of CPTP maps such that each map, $\\mathcal{L}_j$, takes $\\mathcal{B}\\left(\\mathbb{C}^{\\otimes K_j}\\otimes\\mathbb{C}^{\\otimes K_j}\\right)$ to $\\mathcal{B}\\left(\\left(\\mathcal{H}_A\\otimes\\mathcal{H}_B)\\right)^{\\otimes n_j}\\right)$ and belongs to the class $X$.\n\\end{definition}\n\n\\begin{proposition}\n\tThe entanglement cost $E_{c}^{\\mathinner{\\mathrm{PPT}}}(\\rho_{\\lambda})$ and distillable entanglement $E_{d}^{\\mathinner{\\mathrm{PPT}}}(\\rho_{\\lambda})$ of $\\rho_{\\lambda}$ are both equal to $S_{1}(\\lambda)$, the entropy of entanglement of the state.\n\\end{proposition}\n\\begin{proof}\n\t\nIt is clear and well-known that by LOCC operations, both the entanglement cost and distillable entanglement of $\\rho_{\\lambda}$ is $S_{1}(\\lambda)$. By elementary\nresults of Rains' theory of PPT distillation \\cite{rains}, PPT operations\ncannot asymptotically increase the number of EPR pairs. Hence\n$$E_{c}^{\\mathinner{\\mathrm{PPT}}}(\\rho_{\\lambda}) \\geq S_1(\\lambda),\\quad\n E_{d}^{\\mathinner{\\mathrm{PPT}}}(\\rho_{\\lambda}) \\leq S_1(\\lambda).$$\nSince the opposite inequalities are trivial by LOCC $\\subset$ PPT,\nwe conclude that\n$$E_{c}^{\\mathinner{\\mathrm{PPT}}}(\\rho_{\\lambda}) = E_{d}^{\\mathinner{\\mathrm{PPT}}}(\\rho_{\\lambda}) = S_{1}(\\lambda),$$\nthus coinciding with the corresponding LOCC entanglement cost and distillable entanglement: the entropy\nof entanglement.\n\\end{proof}\n\n\\begin{proposition}\n\tThe exact distillable entanglement $E_{xd}^{\\mathinner{\\mathrm{PPT}}}(\\rho_{\\lambda})$ of $\\rho_{\\lambda}$ is given by $S_{\\infty}(\\lambda)$.\n\\end{proposition}\n\\begin{proof}\nWe use a result of Rains~\\cite{rains} on the maximum fidelity\nobtainable from $\\ket{\\psi}$ via PPT operations, to a maximally entangled\nstate of Schmidt rank $K$: by this result, exact transformation (fidelity $1$)\nis possible if there exists an operator $F$ with\n$$\\rho_{\\lambda} \\leq F\\leq\\openone,\\quad -\\frac{1}{K}\\openone\\leq F^\\Gamma\\leq \\frac{1}{K}\\openone.$$\nThus, an upper bound to $K$ is given by\n$$\\max\\left\\{ \\left\\|F^\\Gamma\\right\\|_\\infty^{-1}\\ \\big|\\ \\rho_{\\lambda}\\leq F\\leq\\openone \\right\\}.$$\nWe claim that this is $1\/\\lambda^{\\downarrow}_1$: for assume any\n$F$ as above, and write it $F=\\rho_{\\lambda^{\\downarrow}}+A$, $A\\geq 0$.\n\\begin{equation}\\begin{split}\n \\left\\|F^\\Gamma\\right\\|_\\infty &\\geq \\mathinner{\\mathrm{Tr}}(\\ketbra{11}{11}(\\rho_{\\lambda^{\\downarrow}}^{\\Gamma} + A^{\\Gamma})) \\\\\n &= \\lambda^{\\downarrow}_1 + \\mathinner{\\mathrm{Tr}}(\\ketbra{11}{11}^{\\Gamma} A) \\\\\n& = \\lambda^{\\downarrow}_1 + \\mathinner{\\mathrm{Tr}}(\\ketbra{11}{11} A) \\geq \\lambda^{\\downarrow}_1.\n\\end{split}\\end{equation}\nHence $\\left\\|F^\\Gamma\\right\\|_\\infty\\geq \\lambda^{\\downarrow}_1$, with equality\nachieved for $F=\\rho_{\\lambda}$.\n\\par\nThis gives $E_{xd}^{\\mathinner{\\mathrm{PPT}}}(\\rho_{\\lambda}) \\leq S_\\infty(\\lambda)$.\nThe opposite inequality comes from LOCC operations asymptotically\nachieving this bound~\\cite{morikoshi}.\n\\end{proof}\n\n\\begin{proposition}\\label{exactPPTcost}\nThe entanglement cost $E_{xc}^{\\mathinner{\\mathrm{PPT}}}(\\rho_{\\lambda})$ of $\\rho_{\\lambda}$ is $S_{1\/2}(\\lambda)$.\n\\end{proposition}\n\\begin{proof}\n\tThis is merely an application of the more general result of Audenaert \\emph{et al.} \\cite{ape} to pure states.\n\\end{proof}\n\nAll of these quantities are clearly PPT monotones so they provide a necessary condition for the possibility of a PPT pure state transformation:\n\\begin{proposition}\n\tIf $\\rho_{\\mu} \\stackrel{PPT}{\\rightarrow} \\rho_{\\lambda}$ then\n\t\\begin{align}\n\t\tS_{1\/2}(\\mu) \\geq S_{1\/2}(\\lambda), \\\\\n\t\tS_{1}(\\mu) \\geq S_{1}(\\lambda), \\\\\n\t\tS_{\\infty}(\\mu) \\geq S_{\\infty}(\\lambda).\n\t\\end{align}\n\\end{proposition}\n\nAs Schur concave functions, the R\\'{e}nyi entropies of the Schmidt coefficients at all values of $t \\in [0,\\infty]$ are monotones for LOCC state transformations. Under PPT however, the R\\'{e}nyi entropies for $0 \\leq t < 1\/2$ are \\emph{not} monotones.\n\n\\begin{example}\n \\label{expl:small-alpha}\n Consider a pure state $\\rho_{\\lambda}$ with Schmidt spectrum\n $$\\lambda^{\\downarrow} = \\left( \\frac{1}{20},\\frac{1}{20},\\frac{1}{20},\\frac{4}{20},\\frac{4}{20},\\frac{9}{20} \\right).$$\n \\par\n It is easily verified that $S_{1\/2}(\\lambda)=\\log 5$, and\n indeed, in accordance with the Proposition \\ref{exactPPTcost} the transformation\n $$\\Phi_5^{\\otimes(n+o(n))}\\longrightarrow\\rho_{\\lambda}^{\\otimes n}$$\n is possible by PPT operations for all sufficiently large $n$.\n \\par\n But for $0\\leq \\alpha<1\/2$,\n $$\\log 5 < S_{t}(\\lambda).$$\n Since there are trivial examples of initial states such that the opposite\n inequality is true (e.g.~$\\Phi_6$), the R\\'{e}nyi entropies\n at $0\\leq t<1\/2$ are not PPT monotones.\n \\par\n $S_0(\\lambda)$ is just the logarithm of the Schmidt rank, so, as was noted in \\cite{IshizakaPlenio}, PPT operations can increase the Schmidt rank of pure states, a thing LOCC transformations cannot even do with nonzero probability!\n\\end{example}\n\nIt should be noted that a necessary condition for pure bipartite state transformations by \\emph{separable} operations was recently given by Gheorghiou and Griffiths \\cite{sepTrans}.\n\n\\section{PPT transformations from maximally entangled states.}\\label{sec:frommax}\n\nUnless otherwise stated, the final state is $\\rho_{\\lambda}$, where the Schmidt coefficient vector $\\lambda \\in \\mathbb{R}^d$ is assumed without loss of generality to have no vanishing components (so $d$ is the Schmidt rank of the final state). Since any state with Schmidt rank not greater than $K$ can be produced from $\\Phi_K$ by LOCC, the interesting case for PPT transformations is when the Schmidt rank is increased.\n\n\\begin{lemma}\n\tFor any (pure or mixed) final state $\\rho$, $\\Phi_K \\stackrel{PPT}{\\rightarrow} \\rho$ if and only if the solution to the semidefinite program\n\\begin{equation}\\label{mixedPrimalSDP}\n\t\\min \\{ \\mathinner{\\mathrm{Tr}} \\left(P\\right) | P \\ge 0, -\\left(K-1\\right)P^{\\Gamma} \\leq \\rho^{\\Gamma} \\leq \\left(K+1\\right)P^{\\Gamma} \\},\n\\end{equation}\n\twhere $P$ is an hermitian operator on the same Hilbert space as $\\rho$, is less than or equal to one.\n\\end{lemma}\n\\begin{proof}\n\tThe argument is almost the same as the one given in \\cite{ape}, but we present it here for convenience.\n\tIf there is a PPT map $\\mathcal{L}$ such that $\\mathcal{L}\\left(\\Phi_K\\right) = \\rho$ then the map $\\mathcal{L'}$ made by preceding $\\mathcal{L}$ with the twirl operation \\cite{rains} $\\mathcal{T}$ (which can be implemented by LOCC) is also PPT, and does the same transformation since $\\mathcal{T}\\left(\\Phi_K\\right) = \\Phi_K$. By symmetry it is always possible to write the new map in the form\n\t\\begin{equation}\n\t\t\\mathcal{L'}\\left(\\tau\\right) = F \\mathinner{\\mathrm{Tr}}\\left(\\Phi_K \\tau\\right) + G \\mathinner{\\mathrm{Tr}}\\left(\\left(\\openone - \\Phi_K\\right) \\tau\\right)\n\t\\end{equation}\n\twhere, in order for the map to be CPTP, $F$ and $G$ must be density operators. In order that $\\mathcal{L}'\\left(\\tau\\right) = \\rho$, we require $F = \\rho$, so the desired PPT transformation is possible if and only if there is a density operator $G$ such that the resulting $\\mathcal{L'}$ is PPT. $\\mathcal{L'}$ is PPT preserving if and only if the operator\n\t\\begin{align}\n\t\t\\mathcal{L'}\\left(\\tau ^{\\Gamma}\\right)^{\\Gamma} &= \\rho^{\\Gamma} \\mathinner{\\mathrm{Tr}}\\left(\\Phi_K^{\\Gamma} \\tau\\right) + G^{\\Gamma} \\mathinner{\\mathrm{Tr}}\\left(\\left(\\openone - \\Phi_K\\right)^{\\Gamma} \\tau\\right)\\\\\n\t\t&= \\rho^{\\Gamma} \\mathinner{\\mathrm{Tr}}\\left(\\left(\\mathcal{S} - \\mathcal{A}\\right) \\tau\\right)\/K + G^{\\Gamma} \\left(\\left(\\left(1-1\/K\\right)\\mathcal{S} + \\left(1+1\/K\\right)\\mathcal{A}\\right) \\tau\\right) \\\\\n\t\t&= \\mathinner{\\mathrm{Tr}}\\left(\\mathcal{S} \\tau\\right) \\left(\\rho^{\\Gamma}\/K + \\left(1-1\/K\\right)G^{\\Gamma}\\right) + \\mathinner{\\mathrm{Tr}}\\left(\\mathcal{A} \\tau\\right) \\left(\\left(1+1\/K\\right)G^{\\Gamma} - \\rho^{\\Gamma}\/K\\right)\n\t\\end{align}\n\tis positive semidefinite for all positive semidefinite $\\tau$, where $\\mathcal{S}$ and $\\mathcal{A}$ are the projectors onto the symmetric and anti-symmetric subspaces of the bipartite space. Since $\\mathcal{S}\\mathcal{A} = 0$, this condition holds if and only if $\\rho^{\\Gamma}\/K + \\left(1-1\/K\\right)G^{\\Gamma} \\geq 0$ and $\\left(1+1\/K\\right)G^{\\Gamma} - \\rho^{\\Gamma}\/K \\geq 0$. We also require $G \\geq 0$ and $\\mathinner{\\mathrm{Tr}} G = 1$. The set of operators which satisfy these constraints is precisely the set of feasible points of the SDP \\eref{mixedPrimalSDP} which have trace 1. If $P_0$ is feasible, then points $P_0 + t \\openone$ are also feasible for all $t \\geq 0$, so if an optimal point has trace not greater than one, it ensures the existence of a feasible point which satisfies all the constraints on $G$ (so the transformation is possible); if an optimal point has trace greater than one, then clearly no such point exists and hence the transformation is not possible.\n\t\n\\end{proof}\n\t\n\\begin{proposition}\\label{prop:primal}\n\t\\begin{equation}\n\t\t\\Phi_K \\stackrel{PPT}{\\rightarrow} \\rho_{\\lambda} \\iff T\\left(K;\\lambda\\right) \\leq 1,\n\t\\end{equation}\n\twhere we define $T\\left(K;\\lambda\\right)$ to be the solution to the semidefinite program\n\t\\begin{equation}\\label{primalSDP}\n\t\tT\\left(K;\\lambda\\right) := \\min \\left\\{ \\displaystyle\\sum_{i\\geq j}^{d} s_{ij} + \\displaystyle\\sum_{i > j}^{d} a_{ij} \\bigg| s_{ij} \\geq \\frac{\\sqrt{\\lambda_i \\lambda_j}}{K+1}, a_{ij} \\geq \\frac{\\sqrt{\\lambda_i \\lambda_j}}{K-1}, \\displaystyle\\sum_{i\\geq j} s_{ij} \\sigma_{ij}^{\\Gamma} + \\displaystyle\\sum_{i > j} a_{ij} \\alpha_{ij}^{\\Gamma} \\geq 0 \\right\\},\n\t\\end{equation}\n\twith\n\t\\begin{eqnarray*}\n\t&\\sigma_{i j}& = \\left\\{ \\begin{array}{l}\n\t\t(\\ket{ij} + \\ket{ji}) (\\bra{ij} + \\bra{ji})\/2 \\\\\n\t\t\\ketbra{ii}{ii}\n\t\t \\end{array} \\right. \\begin{array}{c}\n\t\t \\textrm{ when } i \\neq j, \\\\\n\t\t \\textrm{ when } i = j,\n\t\t \\end{array}\\\\\n\t\t &\\alpha_{i j}& = \\left(\\ket{ij} - \\ket{ji}\\right) \\left(\\bra{ij} - \\bra{ji}\\right)\/2,\n\t\\end{eqnarray*}\n\tand $\\{s_{ij} | 1 \\leq j \\leq i \\leq d \\}$, $\\{a_{ij} | 1 \\leq j < i \\leq d \\}$ together constitute $d^2$ real variables.\n\\end{proposition}\n\\begin{proof}\n\\begin{equation}\n\t\\rho_{\\lambda}^{\\Gamma} = \\displaystyle\\sum_{i\\geq j}\\sqrt{\\lambda_i \\lambda_j} \\sigma_{ij} - \\displaystyle\\sum_{i > j}\\sqrt{\\lambda_i \\lambda_j} \\alpha_{ij}.\n\\end{equation}\nLet $\\Pi$ be the projection map on the space of hermitian operators given by\n\\begin{equation}\n\t\\Pi\\left(\\tau\\right) = \\displaystyle\\sum_{i\\geq j}\\sigma_{ij}\\tau\\sigma_{ij} + \\displaystyle\\sum_{i > j}\\alpha_{ij}\\tau\\alpha_{ij}.\n\\end{equation}\nTo show that this map preserves positivity of partial-transpose, we note that\n\\begin{equation}\n\t\\Pi\\left(\\tau^{\\Gamma}\\right)^{\\Gamma} = \\frac{1}{2}\\displaystyle\\sum_{i\\neq j}\\bra{ij}\\tau\\ket{ij}\\left(\\ketbra{ij}{ij} + \\ketbra{ji}{ji}\\right) + \\frac{1}{2}\\displaystyle\\left(\\sum_{i}\\ketbra{ii}{ii}\\right)\\left(\\tau+\\tau^{\\ast}\\right)\\left(\\sum_{j}\\ketbra{jj}{jj}\\right).\n\\end{equation}\nClearly then, $\\Pi$ is positive and preserves positivity of partial transpose. Since $\\rho^{\\Gamma}$ lies in the image of $\\Pi$, if $P$ is a feasible point of the semidefinite program \\eref{mixedPrimalSDP} then $\\Pi\\left(P^{\\Gamma}\\right)^{\\Gamma}$ is also a feasible point. Since $\\Pi$ is also trace reducing, it will not change the optimal value of \\eref{mixedPrimalSDP} if we impose the additional restriction that $P = \\Pi\\left(P^{\\Gamma}\\right)^{\\Gamma}$. This restriction is obeyed if and only if we can write $P$ in the form\n\\begin{equation}\n\tP = \\displaystyle\\sum_{i\\geq j} s_{ij} \\sigma_{ij}^{\\Gamma} + \\displaystyle\\sum_{i > j} a_{ij} \\alpha_{ij}^{\\Gamma}.\n\\end{equation}\nSubstituting this into \\eref{mixedPrimalSDP}, we obtain \\eref{primalSDP}.\n\\end{proof}\n\n\\begin{lemma}\n\t\\begin{equation}\\label{T-upperBound}\n\t\tT\\left(K; \\lambda\\right) \\leq \\left(2^{S_{1\/2}\\left(\\lambda\\right)}-1\\right)\/\\left(K-1\\right).\n\t\\end{equation}\n\\end{lemma}\n\\begin{proof}\n\tThe point\n\t\\begin{equation}\n\t\ts_{ij} = \\sqrt{\\lambda_i \\lambda_j}\/\\left(K-1\\right), a_{ij} = \\sqrt{\\lambda_i\\lambda_j}\/\\left(K-1\\right)\n\t\\end{equation}\n\tis primal feasible since\n\t\\begin{equation}\n\t\t\\displaystyle\\sum_{i} \\lambda_i \\sigma_{ii}^{\\Gamma} + \\displaystyle\\sum_{i > j} \\sqrt{\\lambda_i\\lambda_j} \\left(\\sigma_{ij} + \\alpha_{ij}\\right)^{\\Gamma} = \\displaystyle\\sum_{i} \\lambda_i \\ketbra{ii}{ii} + \\displaystyle\\sum_{i > j} \\sqrt{\\lambda_i\\lambda_j} \\left(\\ketbra{ij}{ij} + \\ketbra{ji}{ji}\\right)\\geq 0\n\t\\end{equation}\n\tand the other inequalities are obviously satisfied. The primal objective function at this point is $2^{S_{1\/2}\\left(\\lambda\\right)}\/\\left(K-1\\right)$.\n\\end{proof}\n\nSince the semidefinite program \\eref{primalSDP} is strictly feasible (take the point $s_{ij} = 2, a_{ij} = 2$, for example), its solution is equal to the solution of the dual SDP \\cite{vbSDP} so we have\n\n\\begin{proposition}\n\t\\begin{align}\n\t\tT\\left(K;\\lambda\\right) = \\max \\left\\{ \\frac{K 2^{S_{1\/2}\\left(\\lambda\\right)} - 1}{K^2 - 1} - \\left(\\sum_{i\\geq j} \\frac{\\mu_{ij}\\sqrt{\\lambda_i \\lambda_j}}{K+1} + \\sum_{i > j} \\frac{t_{ij}\\sqrt{\\lambda_i \\lambda_j}}{K-1} \\right) \\bigg| \\left(\\mu,t\\right) \\in R \\right\\}\\label{dualSDP}\\\\\n\t\tR := \\left\\{\\left(\\mu,t\\right) \\bigg| \\mu_{ij} \\leq 1, t_{ij} \\leq 1, \\sum_{i\\geq j} \\mu_{ij} \\sigma_{ij}^{\\Gamma} + \\sum_{i > j} t_{ij} \\alpha_{ij}^{\\Gamma} \\geq 0 \\right\\}.\n\t\\end{align}\n\tHere $\\{\\mu_{ij} | 1 \\leq j \\leq i \\leq d \\}$, $\\{t_{ij} | 1 \\leq j < i \\leq d \\}$ together constitute $d^2$ real variables.\n\\end{proposition}\n\\begin{proof}\n\tThe semidefinite program dual to \\eref{primalSDP} is\n\t\\begin{align}\n\t\t\\textrm{maximize } \\frac{1}{K+1}\\sum_{i\\geq j}\\sqrt{\\lambda_i \\lambda_j} \\left(1-\\mathinner{\\mathrm{Tr}}\\left(Z \\sigma_{ij}^{\\Gamma}\\right)\\right) + \\frac{1}{K-1}\\sum_{i > j}\\sqrt{\\lambda_i \\lambda_j} \\left(1-\\mathinner{\\mathrm{Tr}}\\left(Z \\alpha_{ij}^{\\Gamma}\\right)\\right)\n\t\\end{align}\n\t\\begin{align}\t\n\t\t\\textrm{subject to } Z &\\geq 0\\\\\n\t\t\\mathinner{\\mathrm{Tr}}\\left(Z \\sigma_{ij}^{\\Gamma}\\right) &\\leq 1\\\\\n\t\t\\mathinner{\\mathrm{Tr}}\\left(Z \\alpha_{ij}^{\\Gamma}\\right) &\\leq 1.\n\t\\end{align}\n\tIf $Z$ is feasible point of this program then so is $\\Pi\\left(Z^{\\Gamma}\\right)^{\\Gamma}$ (where $\\Pi$ is the map defined in the proof of Proposition \\ref{prop:primal}). So the substitutions\n\t\\begin{equation}\n\t\tZ = \\sum_{i\\geq j} \\mu_{ij} \\sigma_{ij}^{\\Gamma} + \\sum_{i > j} t_{ij} \\alpha_{ij}^{\\Gamma}, x_{ij} = 1 - \\mu_{ij}, y_{ij} = 1 - t_{ij}\n\t\\end{equation}\n\tresult in an SDP with the same solution and this is the one given in the proposition.\n\\end{proof}\n\nThe dual objective at any feasible point of the dual semidefinite program \\eref{dualSDP} is a lower bound on $T\\left(K;\\lambda\\right)$, and therefore\n\n\\begin{corollary}\n\t\\begin{equation}\\label{T-lowerBound}\n\t\tT\\left(K; \\lambda\\right) \\geq \\left(K 2^{S_{1\/2}\\left(\\lambda\\right)} - 1\\right)\/\\left(K^2 - 1\\right).\n\t\\end{equation}\n\\end{corollary}\n\\begin{proof}\n\tThe point $\\mu_{ij} = 0, t_{ij} = 0$ is clearly dual feasible.\n\\end{proof}\n\n\\begin{theorem}\\label{t:rank1opt}\n\tThe optimal value of the dual objective that can be attained by a dual feasible point which satisfies the additional constraint that\n\t\\begin{equation}\\label{rankconstraint}\n\t\t\\mathinner{\\mathrm{rank}}\\left(\\displaystyle\\sum_{i\\geq j} \\mu_{ij} \\sigma_{ij}^{\\Gamma} + \\displaystyle\\sum_{i > j} t_{ij} \\alpha_{ij}^{\\Gamma}\\right) = 1,\n\t\\end{equation}\n \tis given by\n\t\\begin{equation}\\label{rankConstrainedOptimum}\n\t\tT_1\\left(K; \\lambda\\right) = \\frac{K 2^{S_{1\/2}\\left(\\lambda\\right)} - 1}{K^2 - 1}+\\frac{K}{\\left(K^2-1\\right)\\left(K+c^{\\ast}-d\\right)}\\left( \\left(\\displaystyle\\sum_{i=1}^{c^{\\ast}}\\sqrt{\\lambda^{\\uparrow}_i}\\right)^2-\\left(K+c^{\\ast}-d\\right)\\displaystyle\\sum_{i=1}^{c^{\\ast}}\\lambda^{\\uparrow}_i\\right),\n\t\\end{equation}\n\twhere $c^{\\ast}$ is the smallest number $c \\in \\{1 + d - K,...,d-1 \\}$ satisfying\n\t\\begin{equation}\\label{c-constraint}\n\t\t\\frac{\\sum_{i=1}^{c}\\sqrt{\\lambda^{\\uparrow}_i}}{K+c-d} \\leq \\sqrt{\\lambda^{\\uparrow}_{c+1}}.\n\t\\end{equation}\n\tIf none of the integers in the range satisfy this relation then $c^{\\ast} = d$.\n\\end{theorem}\n(The proof of this theorem is given in the appendix.)\n\\begin{remark}\n\tClearly $T_1\\left(K; \\lambda\\right) \\leq T\\left(K; \\lambda\\right)$, so a necessary condition for the transformation $\\Phi_K \\stackrel{PPT}{\\rightarrow} \\rho_{\\lambda}$ is $T_1\\left(K; \\lambda\\right) \\leq 1$.\n\\end{remark}\n\n\\begin{corollary}\\label{c:tt1}\n\tIf $\\left(\\displaystyle\\sum_{i=1}^{d-1}\\sqrt{\\lambda^{\\uparrow}_i}\\right)\/\\left(K-1\\right) > \\sqrt{\\lambda^{\\uparrow}_{d}}$ then\n\t\\begin{equation}\n\t\tT\\left(K;\\lambda\\right) = T_1\\left(K;\\lambda\\right) = \\left(2^{S_{1\/2}\\left(\\lambda\\right)}-1\\right)\/\\left(K-1\\right).\n\t\\end{equation}\n\\end{corollary}\n\\begin{proof}\n\tIn this case $c^{\\ast} = d$ and so, $T_1\\left(K;\\lambda\\right) = \\left(2^{S_{1\/2}\\left(\\lambda\\right)}-1\\right)\/\\left(K-1\\right)$ which, by Lemma \\ref{T-upperBound}, is an upper bound on $T\\left(K;\\lambda\\right)$. Since $T_1$ is also a lower bound on $T$, the result follows.\n\\end{proof}\n\n\\begin{corollary}\\label{c:borderline}\n\tIf $S_{1\/2}\\left(\\lambda\\right) = \\log K$ and $d \\geq K$ then the transformation is possible only in the trivial case where the goal state is also a maximally entangled state of rank $K$.\n\\end{corollary}\n\\begin{proof}\n\tThe value $c = d - 1$ satisfies \\eref{c-constraint} provided that $\\lambda^{\\uparrow}_d \\geq 1\/K$ or equivalently if $S_{\\infty}\\left(\\lambda\\right) \\leq \\log K$. Since $S_{t}\\left(\\lambda\\right)$ is a non-increasing function of $t$, this condition is indeed satisfied. Using this value of $c$ in \\eref{rankConstrainedOptimum} yields the lower bound\n\t\\begin{equation}\n\t\tT_1\\left(K; \\lambda\\right) \\geq 1 + \\frac{K\\left(\\sqrt{K \\lambda^{\\uparrow}_d} - 1\\right)^2}{\\left(K^2-1\\right)\\left(K-1\\right)}\n\t\\end{equation}\n\twhich is clearly greater than $1$ (implying the impossibility of the transformation) except where $S_{\\infty}\\left(\\lambda\\right) = \\log K $ which (together with $S_{1\/2}\\left(\\lambda\\right) = \\log K$) implies $\\lambda$ is the uniform distribution of size $K$, so the goal state is a maximally entangled state of rank $K$.\n\\end{proof}\n\n\\begin{proposition}\\label{p:rank3}\n\tIn the case where the goal state has Schmidt rank three, $T\\left(2; \\lambda\\right) = T_1\\left(2; \\lambda\\right)$.\n\\end{proposition}\n\\begin{proof}\n\tTo simplify notation we shall here assume that $\\lambda_i = \\lambda^{\\uparrow}_i$. If $\\sqrt{\\lambda_1} + \\sqrt{\\lambda_2} > \\sqrt{\\lambda_3}$ then we can apply Corollary \\ref{c:tt1} and we're done, so we assume that $\\sqrt{\\lambda_1} + \\sqrt{\\lambda_2} \\leq \\sqrt{\\lambda_3}$. In this case\n\t\\begin{equation}\n\t\tT_1 = \\left(1 + 4\\left(\\sqrt{\\lambda_3 \\lambda_2} + \\sqrt{\\lambda_3 \\lambda_1}\\right) + 8\\sqrt{\\lambda_1 \\lambda_2 }\\right)\/3.\n\t\\end{equation}\n\tWe shall show that $T$ is the same by constructing a primal optimal solution. Let\n\t\\begin{eqnarray}\n\t\ts^{\\ast}_{11} = \\frac{1}{3}\\lambda_1 + \\frac{4}{9}\\sqrt{\\lambda_1 \\lambda_2}, s^{\\ast}_{22} = \\frac{1}{3}\\lambda_2 + \\frac{4}{9}\\sqrt{\\lambda_1 \\lambda_2}, s^{\\ast}_{33} = \\frac{1}{3}\\lambda_3, \\\\\n\t\ts^{\\ast}_{12} = \\left(\\frac{1}{3} + \\frac{4}{9}\\right)\\sqrt{\\lambda_1 \\lambda_2}, s^{\\ast}_{13} = \\frac{1}{3}\\sqrt{\\lambda_1 \\lambda_3}, s^{\\ast}_{23} = \\frac{1}{3}\\sqrt{\\lambda_2 \\lambda_3}, \\\\\n\t\ta^{\\ast}_{ij} = \\sqrt{\\lambda_i \\lambda_j}\n\t\\end{eqnarray}\n\t\\begin{equation}\n\t\\displaystyle\\sum_{i\\geq j} s^{\\ast}_{ij} \\sigma_{ij}^{\\Gamma} + \\displaystyle\\sum_{i > j} a^{\\ast}_{ij} \\alpha_{ij}^{\\Gamma} = P_1\\oplus P_2\n\t\\end{equation}\n\t\\begin{eqnarray*}\n\t P_1 = \\frac{2}{3} \\sum_{i \\neq j} \\sqrt{\\lambda_i\n\t \\lambda_j} \\ketbra{ij}{ij} + \\frac{2}{9} \\left(\\ketbra{23}{23} + \\ketbra{32}{32}\\right) \\geq 0\n\t\\end{eqnarray*}\n\tWritten as a matrix in the $\\{\\ketbra{ii}{jj}\\}$ basis,\n\t\\begin{eqnarray*}\n\t P_2 = \\frac{1}{9} \\left(\\begin{array}{ccc}\n\t 3 \\lambda_1 + 4 \\sqrt{\\lambda_2 \\lambda_3} & - 3 \\sqrt{\\lambda_1 \\lambda_2} & - 3 \\sqrt{\\lambda_1\n\t \\lambda_3}\\\\\n\t - 3 \\sqrt{\\lambda_2 \\lambda_1} & 3 \\lambda_2 + 4 \\sqrt{\\lambda_2\n\t \\lambda_3} & - \\sqrt{\\lambda_2 \\lambda_3}\\\\\n\t - 3 \\sqrt{\\lambda_3 \\lambda_1} & - \\sqrt{\\lambda_3 \\lambda_2} & 3\n\t \\lambda_3\n\t \\end{array}\\right) & & \n\t\\end{eqnarray*}\n\twhich can be seen to be positive semidefinite by the Sylvester criterion. Therefore $P_1 \\oplus P_2 \\geq 0$, and since the other primal constraints are clearly satisfied, the point $\\left(s^{\\ast},a^{\\ast}\\right)$ is primal feasible so\n\t\\begin{equation}\n\t\tT_1\\left(2;\\lambda\\right) \\leq T\\left(2;\\lambda\\right) \\leq \\displaystyle\\sum_{i\\geq j} s^{\\ast}_{ij} + \\displaystyle\\sum_{i > j} a^{\\ast}_{ij} = \\left(1 + 4\\left(\\sqrt{\\lambda_3 \\lambda_2} + \\sqrt{\\lambda_3 \\lambda_1}\\right) + 8\\sqrt{\\lambda_1 \\lambda_2 }\\right)\/3 = T_1\\left(2;\\lambda\\right).\n\t\\end{equation}\n\tTherefore $T\\left(2;\\lambda\\right) = T_1\\left(2;\\lambda\\right)$.\n\\end{proof}\nIt follows directly from this proposition that:\n\\begin{theorem}\\label{t:rank3}\n\tA pure state of Schmidt rank three can be produced from $\\Phi_2$ (an EPR pair) by PPT operations if and only if its Schmidt coefficients obey\n\t\\begin{equation}\n\t\t2\\left( \\sqrt{\\lambda^{\\uparrow}_3 \\lambda^{\\uparrow}_2} + \\sqrt{\\lambda^{\\uparrow}_3 \\lambda^{\\uparrow}_1} \\right) + 4\\sqrt{\\lambda^{\\uparrow}_1\\lambda^{\\uparrow}_2} \\leq 1.\n\t\\end{equation}\n\\end{theorem}\n\nThis fact, and some suggestive numerical evidence, leads us to make the following conjecture\n\\begin{conjecture}\\label{conj:rankIs1}\n\tIn the dual program \\eref{dualSDP}, the optimal value of the dual objective function is always attained by a point satisfying the rank constraint \\eref{rankconstraint} and as a consequence $T\\left(K; \\lambda\\right) = T_1\\left(K; \\lambda\\right)$.\n\\end{conjecture}\n\n\\section{Catalysis}\\label{sec:cat}\n\nNielsen's theorem was used to show that a phenomenon analogous to chemical catalysis can occur in LOCC entanglement transformation \\cite{catalysis}. That is, there exist pairs of states $\\rho_{\\lambda}, \\rho_{\\mu}$ such that $\\rho_{\\lambda} \\stackrel{LOCC}{\\nleftrightarrow} \\rho_{\\mu}$ but where there is a third state $\\rho_{\\xi}$ such that $\\rho_{\\lambda}\\otimes\\rho_{\\xi} \\stackrel{LOCC}{\\rightarrow} \\rho_{\\mu}\\otimes\\rho_{\\xi}$.\n\nA necessary and sufficient condition for the existence of a deterministic catalytic transformation for bipartite pure states was recently given in papers by Turgut \\cite{turgut} and Klimesh \\cite{klimesh}:\n\n\\begin{theorem}\\label{t:locccatcond}\nGiven two pure bipartite states $\\rho_{\\lambda}$ and $\\rho_{\\mu}$, where $\\lambda^{\\uparrow} \\neq \\mu^{\\uparrow}$ and $\\lambda$ and $\\mu$ don't both have components equal to zero, there exists a pure state $\\rho_{\\xi}$ such that $\\rho_{\\lambda}\\otimes\\rho_{\\xi} \\stackrel{LOCC}{\\rightarrow} \\rho_{\\mu}\\otimes\\rho_{\\xi}$ if and only if the following conditions are satisfied\n\\begin{align}\n\tS_{t}\\left(\\lambda\\right) &> S_{t}\\left(\\mu\\right) \\textrm{ for } t \\in \\left(0, \\infty\\right), \\\\\t\n\tf_{t}\\left(\\lambda\\right) &> f_{t}\\left(\\mu\\right) \\textrm{ for } t \\in \\left(-\\infty, 0\\right],\n\\end{align}\nwhere\n\\begin{equation}\n\tf_t\\left(\\lambda\\right) := \\left\\{ \\begin{array}{l}\n\t\t\\frac{1}{t - 1}\\log \\left( \\sum_{i=1}^{d} \\lambda_i^t \\right)\\\\\n\t\t\\sum_i \\log \\lambda_i \\\\\n\t\t-\\infty\n\t\t \\end{array} \\right. \\begin{array}{l}\n\t\t \\textrm{ when } t > 0 \\textrm{ and } \\lambda_i \\neq 0 \\textrm{ for all } i \\in \\{1,...,d\\}, \\\\\n\t\t\t\\textrm{ when } t = 0 \\textrm{ and } \\lambda_i \\neq 0 \\textrm{ for all } i \\in \\{1,...,d\\}, \\\\\n\t\t \\textrm{ otherwise. }\n\t\t \\end{array}\n\\end{equation}\n\\end{theorem}\n\nUsing the results established in the last section, we can show that catalysis can also occur under PPT operations.\n\n\\begin{theorem}\\label{t:catcond}\n\t$\\Phi_C\\otimes\\Phi_K \\stackrel{PPT}{\\rightarrow} \\Phi_C\\otimes\\rho_{\\lambda}$ (where $\\rho_\\lambda \\neq \\Phi_K$) for some sufficiently large value of $C$ if and only if $S_{1\/2}\\left(\\lambda\\right) < \\log K$.\n\\end{theorem}\n\\begin{proof}\n\tFor a given value of $C$, the transformation is possible if and only if $T\\left(KC;\\lambda\\otimes U_C\\right) \\leq 1$, where $U_C$ denotes the uniform probability distribution vector with $C$ elements.\n\tAs $C$ tends to infinity, both the upper bound \\eref{T-upperBound} and lower bound \\eref{T-lowerBound} on $T\\left(KC;\\lambda\\otimes U_C\\right)$ tend to $2^{S_{1\/2}\\left(\\lambda\\right)}\/K$. Therefore, if $S_{1\/2}\\left(\\lambda\\right) < \\log K$ then for some sufficiently large $C$, $T\\left(KC;\\lambda\\otimes U_C\\right) \\leq 1$ so the corresponding transformation is possible. That the condition is necessary follows from Proposition \\ref{exactPPTcost} ($S_{1\/2}$ cannot increase in \\emph{any} PPT pure state transformation and the R\\'{e}nyi entropies are additive) and Corollary \\ref{c:borderline} (deals with the case where $S_{1\/2}$ stays the same).\n\\end{proof}\n\nIn the case where the goal state has Schmidt rank three, the states satisfying\n\\begin{equation}\n\t2\\left (\\sqrt{\\lambda^{\\uparrow}_3 \\lambda^{\\uparrow}_2} + \\sqrt{\\lambda^{\\uparrow}_3 \\lambda^{\\uparrow}_1} + \\sqrt{\\lambda^{\\uparrow}_1 \\lambda^{\\uparrow}_2 } \\right) \\leq 1\n\\end{equation}\nare exactly those that can be reached from $\\Phi_2$ by PPT operations when maximally entangled catalysts of arbitrarily high rank are available, according to Theorem \\ref{t:catcond}. This is a strict superset of those Schmidt rank three states which can be obtained from $\\Phi_2$ without a catalyst (see Theorem \\ref{t:rank3}). These regions are illustrated on one cell of the simplex of Schmidt coefficient vectors in Fig. \\ref{fig:allowed-regions}.\n\t\n\\begin{figure}\n\\includegraphics[scale=0.5]{regionsDiag.pdf}\n\\caption{Pure states of Schmidt rank three accessible deterministically from a single EPR pair by PPT and by PPT assisted by a maximally entangled catalyst.}\\label{fig:allowed-regions}\n\\end{figure}\n\n\\section{Conclusions}\\label{sec:conc}\n\nWe have provided a necessary condition for the exact preparation of a pure bipartite state from a maximally entangled state by PPT operations in terms of the Schmidt coefficients of the final state. We conjecture that this condition is also sufficient and have shown that this is true when the final state has Schmidt rank three. We have demonstrated that the phenomenon of catalysis occurs in the context of PPT operations. A notable difference from LOCC catalysis is that maximally entangled states can act as catalysts under PPT operations (this is impossible with LOCC). In the case where both the initial state and the catalyst are maximally entangled, we have given a necessary and sufficient condition for the production of a pure state: the R\\'{e}nyi entropy at $1\/2$ of the Scmidt coefficient vector must decrease, unless the final state is the same as the initial state. To give a direction for future work -- comparing Theorem \\ref{t:catcond} and the observations on PPT monotones given in Section \\ref{sec:general} with Theorem \\ref{t:locccatcond}, we make the following conjecture:\n\n\\begin{conjecture}\n\tWhen $\\lambda^{\\uparrow} \\neq \\mu^{\\uparrow}$, there exists a catalyst state $\\rho_{\\xi}$ such that $\\rho_{\\xi}\\otimes\\rho_{\\lambda} \\stackrel{PPT}{\\rightarrow} \\rho_{\\xi}\\otimes\\rho_{\\mu}$ if and only if\n\t\\begin{equation}\n\tS_{t}\\left(\\lambda\\right) > S_{t}\\left(\\mu\\right), \\forall t \\in [1\/2,\\infty).\n\t\\end{equation}\n\\end{conjecture}\n\nIt would also be desirable to determine the validity of Conjecture \\ref{conj:rankIs1}.\n\n\\begin{acknowledgments}\nWM acknowledges support from the U.K. EPSRC. \nAW is supported by the U.K. EPSRC (project ``QIP IRC'' and an Advanced\nResearch Fellowship), by a Royal Society Wolfson Merit Award, and\nthe EC, IP ``QAP''. The Centre for Quantum Technologies is funded by the\nSingapore Ministry of Education and the National Research Foundation as\npart of the Research Centres of Excellence programme.\nWe thank Armin Uhlmann and Jens Eisert for early discussions on the topic of this\npaper, and Richard Low for his comments on presentation.\n\\end{acknowledgments}\n\n\t\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\\label{s-in}\n\nThe classical Pohozaev-Schoen identity \\cite{Schoen} lies within a larger set of geometric identities which rely on conservation principles arising from symmetries of specific variational problems, as has been noted and explained in \\cite{Gover} (see also \\cite{allan} for another variant, in manifolds with boundary, of this useful identity). In this work, the authors encapsulate several geometric identities which have had important impact in geometric analysis within a single identity derived from such symmetry principles. It is important to note that these type of geometric identities have been established for compact manifolds. Taking into account how useful these conservation principles have proved to be when analysing geometric problems, we intend to present a version of the Pohozaev-Schoen identity established in \\cite{allan} for asymptotically euclidean (AE) manifolds. These manifolds are complete non-compact manifolds with a particular simple structure at infinity. In general relativity, for example, we often need to consider initial data problems on non-compact manifolds, with natural restrictions on the asymptotic geometry (see, for instance, \\cite{CB2},\\cite{maxwell} and references therein). The positive mass theorem and the Penrose inequality are example of such situations (see \\cite{Bartnik},\\cite{Huisken},\\cite{Schoen-Yau1},\\cite{Schoen-Yau},\\cite{Witten}). In fact, geometric problems on this scenario have received plenty of attention since these structures play a central role in general relativity, serving as a model for initial data of isolated gravitational systems. Thus, plenty of analytic tools have been developed in this scenario. In this direction, our aim is to prove the validity of a version of the Pohozaev-Schoen identity in this context, which is sufficiently powerful to be useful when analysing a wide variety of geometric problems on AE manifolds. Explicitly, we will prove the following theorem.\\footnote{See Section 2 for the precise definition of the functional spaces involved.}\n\n\\begin{thm*}\nLet $(M,g)$ be a $H_{s,\\delta}$-asymptotically euclidean manifold, with $s>\\frac{n}{2}+1$ and $\\delta>-\\frac{n}{2}$. Also, let $B$ be a symmetric $(0,2)$-tensor field, satisfying $\\mathrm{div}_gB=0$, and let $X$ be a vector field on $M$. Suppose $X\\in C^1$ is bounded with $\\nabla X\\in L^2$. Furthermore, suppose that $X\\in L^2_{-(\\rho+1)}$ and $B\\in H_{s+1,\\rho}$, with $\\rho\\geq 0$. Then, the following equality holds:\n\\begin{align}\n\\int_M\\langle \\overset{\\circ}{B},\\pounds_X g \\rangle\\mu_g = \\frac{2}{n} \\int_M X(\\mathrm{tr}_gB)\\mu_g + 2 \\int_{\\partial M}\\overset{\\circ}{B}(X,\\nu)\\mu_{\\partial M}\n\\end{align}\n\\end{thm*}\n\n\\bigskip\nWe will employ this version of the generalized Pohozaev-Schoen identity to analyse different geometric problems, such as generalized solitons on AE manifolds. These soliton-type equations arise as a generalization of Ricci almost solitons, as proposed by Pigola \\textit{et al.} \\cite{Pigola}. Also these type of equations were analysed in the \\textit{extrinsic} context by Al\\'ias \\textit{et al.} in \\cite{ALR}. During the analysis of these generalized solitons, we will prove the following rigidity theorem.\n\n\\begin{thm*}\nLet $(M^n,g)$ be an asymptotically euclidean manifold without boundary, $n\\geq 3$, where $g$ satisfies the hypotheses of the above theorem, and suppose that the tensor field $B\\in H_{s+1,0}$, $s>\\frac{n}{2}$, is $g$-divergence-free. Furthermore, let $X\\in H_{s+2,-1}$ and $\\mathrm{tr}_gB=0$. Then, under these assumptions, any $B$-generalized soliton is $B$-flat.\n\\end{thm*}\n\nFrom this theorem, we will extract some interesting geometric consequences. For instance, we will show that any AE Ricci almost soliton with zero scalar curvature is trivial, \\textit{i.e}, is isometric to the euclidean flat space. \n\n\\bigskip\nAnother important result, where we will make use of the generalized Pohozaev-Schoen identity, is the following almost-Schur theorem for AE manifolds (see in Section \\ref{applications} a review and motivation for this type of inequalities): \n\n\\begin{thm*}\nLet $(M^n,g)$ is an $H_{s+3,\\delta}$-asymptotically euclidean manifold without boundary, with $n\\geq 3$, $s>\\frac{n}{2}$, $\\delta>-\\frac{n}{2}$ and $\\delta\\geq -2$. Now, let $B$ be a symmetric $(0,2)$-tensor field such that $\\mathrm{div}_gB=0$ and $B\\in H_{s+1,\\rho}$, with $\\rho>-\\frac{n}{2} + 2$, furthermore, suppose that $\\rho\\geq 0$, and denote its trace by $b\\doteq \\mathrm{tr}_gB$. Then:\\\\\nI) there is a constant $C$ independent of the specific choice of such tensor $B$, such that the following inequality holds\n\\begin{align}\n||b||_{L^2}\\leq n\\left( \\frac{n-1}{n} + C||\\mathrm{Ric}_g||_{C^0_2}\\right)^{\\frac{1}{2}} ||\\overset{\\circ}{B}||_{L^2},\n\\end{align}\nII) If $\\mathrm{Ric}_g$ is non-negative, \\textit{i.e}, $\\mathrm{Ric}_g(X,X)\\geq 0$ $\\forall$ $X\\in\\Gamma(TM)$, then\n\\begin{align}\n||b||_{L^2}\\leq \\sqrt{n(n-1)} ||\\overset{\\circ}{B}||_{L^2}.\n\\end{align}\n\\end{thm*}\n\n\\bigskip\nNotice that the hypotheses of the theorem split depending on whether $n=3,4$ or $n\\geq 5$. In the first case, the hypotheses on the weights $\\delta$ and $\\rho$ are $\\delta>-\\frac{n}{2}$ and $\\rho>-\\frac{n}{2}+2$, while, if $n\\geq 5$, we get $\\delta\\geq -2$ and $\\rho\\geq 0$. One reason for these seemingly odd behaviour is that we need, as an underlying hypotheses, both $\\mathrm{Ric}_g$ and $B$ in $L^2$. In dimensions $3$ and $4$ this behaviour of the Ricci tensor is guaranteed by the condition $\\delta>-\\frac{n}{2}$. In contrast, for $n\\geq 5$, this is not the case, and the condition $\\delta\\geq -2$ guarantees the desired asymptotic behaviour. The weight $\\rho$ is subtlety tied to $\\delta$ via the solutions of a PDE (see the proof of Theorem \\ref{B-AS}).\n\n\nFrom this general result, we will extract as corollaries an almost-Schur inequality for the scalar curvature, analogous to the one proved by De Lellis-Topping in \\cite{Topping}; for $Q$-curvature, analogous to \\cite{chinos}, and for the mean curvature of AE-hypersurfaces in Ricci-flat spaces, analogous to \\cite{Cheng1}. Some of these results, in the compact case, where summarized and put together in \\cite{Cheng2}.\n\n\\bigskip\nFinally, we will analyse static potentials on AE manifolds. These structures have recieved plenty of interest and some important results, together with applications, can be consulted in J. Corvino's influential paper \\cite{Corvino}. Strong rigidity of these structures has been observed as a consequence of integral geometric identities, for instance, in \\cite{allan} by Barbosa \\textit{et al.} and in \\cite{Miao} by Miao-Tam for compact and AE manifolds respectively. In this context, we will show the validity of the following result.\n\n\\begin{thm*}\nSuppose that $(M^n,g)$ is a $H_{s+3,\\delta}$-AE manifold with $n\\geq 3$, $s>\\frac{n}{2}$ and $\\delta>-1$, which admits a non-negative static potential $f$. Furthermore, suppose that $\\mathrm{Ric}_g\\in H_{s+1,\\rho}$, for some $\\rho>\\frac{n}{2}-1$, and that $\\partial M=f^{-1}(0)$ consists of $N$-closed connected components, labelled by $\\{\\Sigma_i \\}_{i=1}^{N}$. Then, it follows that \n\\begin{align}\n\\int_Mf|\\mathrm{Ric}_g|^2_g\\mu_g = \\frac{1}{2}\\sum_{i=i}^Nc_i\\int_{\\Sigma_i}R_{h_i}\\mu_{\\partial M},\n\\end{align}\nwhere the constants $c_i=|\\nabla f|_{\\Sigma_i}$. In particular, if $\\partial M=\\emptyset$, then $(M^n,g)$ is isometric to $(\\mathbb{R}^n,e)$, where $e$ is the euclidean metric.\n\\end{thm*}\n\nThis theorem, in dimension three, was proved in \\cite{Miao}. We will show how their result has a natural (although non-trivial) extension to general dimensions. Furthermore, we will show how this result can be used to give very simple characterizations for the admissible topologies for the event horizon of static black holes. The characterization we will provide is not new, although it is derived using much simpler techniques than the ones used to prove more general results, such as the ones appearing in \\cite{EGP} and \\cite{Hawking}. \n\n\\bigskip\nWith the above in mind, the organization of the paper will be as follows. We will first review some of the analytic tools needed for the core of the paper. Then, under sufficient conditions, we will establish the validity of the Pohozaev-Schoen identity on asymptotically euclidean manifolds. Finally, we will go into the applications, where we will show how some rigidity results of generalized solitons can be derived as a straightforward consequence of this identity. Furthermore, a generalized almost-Schur-type inequality, in this non-compact setting, can also be obtained using these tools. As our last application, we will show how some identities which are related with rigidity of static potentials on asymptotically euclidean manifolds, also follow directly from these principles.\n\n\\section{AE Manifolds}\n\nIn this section, the idea will be to present the main definitions and properties of AE manifolds that we will use in the subsequent sections. The analysis of these structures has been developed along the years, including the seminal works \\cite{Bartnik}, \\cite{Cantor}, \\cite{CB1}, \\cite{Maxwell-Dilts}, \\cite{maxwell}, \\cite{maxwell1}. We are following the classical notations and results established in \\cite{CB1}, where the detailed proofs can be consulted.\n\n\\begin{defn}\nA $n$-dimensional smooth Riemannian manifold $(M,e)$ is called euclidean at infinity if there exits a compact set $K$ such that $M\\backslash K$ is the disjoint union of a finite number of open sets $U_i$, such that each $(U_i,e)$ is isometric to the exterior of an open ball in the Euclidean space. \n\\end{defn}\n\nOn manifolds euclidean at infinity, we define $d=d(x,p)$ the distance in the Riemannian metric $e$ of an arbitrary point $x$ to a fixed point $p$. We will typically omit the dependence on $(x,p)$. \n\n\\begin{defn}\nA weighted Sobolev space $H_{s,\\delta}$, with $s$ a nonnegative integer and $\\delta\\in\\mathbb{R}$, is a space of tensor fields $u$ of some given type on the manifold $(M,e)$ Euclidean at infinity with generalized derivatives of order up to $s$ in the metric $e$ such that $D^mu(1 + d^2)^{\\frac{1}{2}(m+\\delta)}\\in L^2$, for $0\\leq m\\leq s$. It is a Banach space with norm\n\\begin{align}\\label{norm}\n||u||^{2}_{H_{s,\\delta}}\\doteq \\sum_{0\\leq m\\leq s}\\int_M|D^mu|_e^2(1 + d^2)^{(m+\\delta)}d\\mu_{e}\n\\end{align}\nwhere $D$ represents the $e$-covariant derivative and $\\mu_e$ the Riemannian volume form associated with $e$.\n\\end{defn}\n\n\\begin{remark}\nIt is important to note that the space $C^{\\infty}_0(M,E)$ of compactly supported tensor fields of some given type is dense in $H_{s,\\delta}$ for all $s\\in\\mathbb{N}$ and $\\delta\\in\\mathbb{R}$. \n\\end{remark}\n\n\n\\begin{defn}\\label{AE-def}\nWe will say that a Riemannian metric $g$ on $M$ is asymptotically euclidean if $g-e\\in H_{s,\\delta}$, $s>\\frac{n}{2}$ and $\\delta>-\\frac{n}{2}$.\n\\end{defn}\n\nThese weighted spaces share several properties of the usual Sobolev spaces. For instance, we have the continuous embedding:\n\n\\begin{lemma}\\label{sobolevembedding}\nLet $(M, e)$ be a manifold euclidean at infinity. The following inclusion holds and is continuous:\n\\begin{align*}\nH_{s,\\delta}\\subset C^{s'}_{\\delta'}\n\\end{align*}\nif $s' \\frac{1}{2}$, we have a continuous trace map\\footnote{Here we are appealing to Sobolev spaces defined for non-integer $s\\in\\mathbb{R}$, which we regard as a standard tool. See, for instance, \\cite{Taylor} for the details.}\n\\begin{align*}\n\\tau: H_{s}(\\Omega) \\mapsto H_{s-\\frac{1}{2}}(\\partial\\Omega).\n\\end{align*}\n\nIn the case that $u$ is actually a section of a vector bundle $E\\xrightarrow{\\pi} \\Omega$, we get an analogous result. To see this, notice that the use of boundary charts trivializing $E$ naturally gives us an induced vector bundle $\\tilde{E}$ over $\\partial\\Omega$ with the same typical fibre as $E$, whose vector bundle structure comes from the boundary coordinate charts, which come from the coordinate systems of $\\Omega$ adapted to the boundary that trivialize $E$. Clearly, we can both extend the sections of $\\tilde{E}$ to section of $E$, and, vice versa, restrict sections of $E$ to sections of $\\tilde{E}$. Hence, we will drop the tilde from $\\tilde{E}$, and take this setting as implicit. Now, notice that the restriction of a section of $E$ to $\\partial\\Omega$ will produce a map $\\tau u: \\partial\\Omega\\mapsto E$. Furthermore, by localizing $u$ in a fashion analogous to what was done prior to introducing the norm (\\ref{locsob}), we get that $u\\in H_{s}(\\Omega,E)$ iff given a partition of unity $\\{\\alpha_i \\}$ subordinate to a coordinate cover $\\{U_i,\\varphi_i \\}$, the localized fields $\\alpha_i u \\in H_s(U_i,E)$. We can then use boundary charts to trace these localized fields, and get, $\\forall$ $s>\\frac{1}{2}$, a continuous trace map \\footnote{The interested reader can find the details on \\cite{Palais}}\n\\begin{align*}\n\\tau:H_{s}(\\Omega,E) \\mapsto H_{s-\\frac{1}{2}}(\\partial\\Omega,E).\n\\end{align*}\n\nIn the case of $AE$-manifolds, since the trace map only involves the compact core $K\\subset\\subset M$, from the above and a partition of unity argument, we get that, for $s>\\frac{1}{2}$, we have continuous trace map\na continuous trace map\n\\begin{align}\n\\tau:H_{s,\\delta}(M,E) \\mapsto H_{s-\\frac{1}{2}}(\\partial M,E).\n\\end{align}\n\n\\begin{remark}\nIn the subsequent sections, in the notation $H_{s,\\delta}(M,E)$, whenever the vector bundle is implicit from the context, we will simply write $H_{s,\\delta}$. The same will be done with other functional spaces.\n\\end{remark}\n\n\\begin{remark}\nWe would like to point out that the definition of AE Riemannian manifold provided in Definition \\ref{AE-def} follows the same lines as the definition given in \\cite{CB1}-\\cite{CB2} and \\cite{Bartnik}, among others. In particular, the definition adopted by Bartnik in \\cite{Bartnik} is related to ours by a simple shift in the weight parameter. Explicitly, $\\delta_{b}=-(\\delta + \\frac{n}{2})$, where $\\delta_b$ is the weighting parameter used in the definition of the functional spaces in \\cite{Bartnik}. Nevertheless, analogous definitions can be given without invoking weighted Sobolev spaces. For instance, it is quite usual to define an AE metric on a manifold euclidean at infinity by its behaviour in each end in coordinate systems. That is, demanding $g$, near infinity, to satisfy a condition of the form\n\\begin{align}\\label{AE-def2}\ng_{ij}=\\delta_{ij} + o_{k}(|x|^{-\\tau}),\n\\end{align}\nwhere $o_{k}(|x|^{-\\tau})$ implies that $\\partial^{l}g_{ij}=o(|x|^{-\\tau-l})$ for all $1\\leq l \\leq k$ and $\\tau>0$. It should be completely clear that the condition expressed in Definition \\ref{AE-def} controls the behaviour of $g$ and its derivatives at infinity and implies that the metric satisfies a conditions of the form of (\\ref{AE-def2}). Also, similar definitions could be given by means of weighted $C^k$-spaces and H\\\"older spaces. All of these definitions are very easily related by means of embedding theorems and analysis of the behaviour of the field at infinity, and the precise choice relies on the specific problems to be treated and which of them seems to be more transparently tailored to deal with such problems.\n\\end{remark}\n\n\\section{Pohozaev-Schoen identity on AE manifolds}\n\nThe idea now will be to try to deduce the integral identity in the context of AE manifolds. We will consider that such manifolds $M$ may have compact boundary $\\partial M$ as described in the previous section.\n\n\n\n\\begin{thm}\\label{thm1}\nLet $(M,g)$ be a $H_{s,\\delta}$-asymptotically euclidean manifold, with $s>\\frac{n}{2}+1$ and $\\delta>-\\frac{n}{2}$. Also, let $B$ be a symmetric $(0,2)$-tensor field, satisfying $\\mathrm{div}_gB=0$, and let $X$ be a vector field on $M$. Suppose $X\\in C^1$ is bounded with $\\nabla X\\in L^2$. Furthermore, suppose that $X\\in L^2_{-(\\rho+1)}$ and $B\\in H_{s+1,\\rho}$, with $\\rho\\geq 0$. Then, the following equality holds:\n\\begin{align}\\label{PS}\n\\int_M\\langle \\overset{\\circ}{B},\\pounds_X g \\rangle\\mu_g = \\frac{2}{n} \\int_M X(\\mathrm{tr}_gB)\\mu_g + 2 \\int_{\\partial M}\\overset{\\circ}{B}(X,\\nu)\\mu_{\\partial M}\n\\end{align}\n\\end{thm}\n\n\n\\begin{proof}\n\nWith a slight abuse of notation, making contact with standard notation for differential forms, we will denote the contraction of $B$ and $X$ by $i_XB\\doteq B(X,\\cdot)$. Now, if we consider $B\\in C^{\\infty}_0$ and $X$ satisfying the hypotheses described above, then the following is standard:\n\\begin{align}\\label{PS1}\n\\frac{1}{2}\\int_M\\langle B,\\pounds_{X}g\\rangle\\mu_g=\\int_M\\mathrm{div}_g(i_XB)\\mu_g=\\int_{\\partial M}B(X,\\nu)\\mu_{\\partial M},\n\\end{align}\nwhere $\\nu$ denotes the unit normal of $\\partial M$. Now, consider $\\{B_n\\}_{n=1}^{\\infty}\\subset C^{\\infty}_0(M)$, such that \n\\begin{align*}\nB_n&\\xrightarrow{H_{s+1,\\rho}} B.\n\\end{align*}\nThen, we have the following:\n\\begin{align*}\n\\Big\\vert \\int_M(\\langle B,\\pounds_{X}g\\rangle-\\langle B_n,\\pounds_{X}g\\rangle)\\mu_g\\Big\\vert&\\leq \\int_M|\\langle B-B_n,\\pounds_{X}g\\rangle|\\mu_g,\\\\\n&\\leq \\int_M|B-B_n|_g|\\pounds_{X}g|_g\\mu_g,\\\\\n&\\leq ||B-B_n||_{L^2}||\\pounds_{X}g||_{L^2}\\xrightarrow[n\\rightarrow\\infty]{} 0,\n\\end{align*}\nwhere we have made use of the fact that, since $\\nabla X\\in L^2$, so does $\\pounds_{X}g$. All this gives us \n\\begin{align}\\label{PS2}\n\\frac{1}{2}\\int_M\\langle B,\\pounds_{X}g\\rangle\\mu_g=\\lim_{n\\rightarrow\\infty}\\int_{\\partial M}B_n(X,\\nu)\\mu_{\\partial M}.\n\\end{align}\nNow, a similar argument gives us\n\\begin{align*}\n\\Big\\vert\\int_{\\partial M}(B-B_n)(X,\\nu)\\mu_{\\partial M}\\Big\\vert\\lesssim \\int_{\\partial M} |B-B_n|_g|X|_g\\mu_{\\partial M}\n\\end{align*}\nwhere we have used that, given two tensor fields $A,B$ on $M$ then the following pointwise estimate holds $|C(A\\otimes B)|_g\\leq K(n,d_1,d_2)|A|_g|B|_g$, where $C$ denotes an arbitrary contraction and $K$ is a constant depending only on the dimension $n$ of $M$ and the dimensions $d_1$ and $d_2$ of the fibres. Notice that, since we have a continuous trace $\\tau: H_{s+1,\\rho}(M)\\mapsto H_{s}(\\partial M)$, then $\\tau(B-B_n)\\xrightarrow[n\\rightarrow \\infty]{} 0$ in $L^2(\\partial M)$. Hence, from the above, we get\n\\begin{align*}\n\\Big\\vert\\int_{\\partial M}(B-B_n)(X,\\nu)\\mu_{\\partial M}\\Big\\vert\\lesssim ||B-B_n||_{L^2(\\partial M)} ||X||_{L^2(\\partial M)}\\xrightarrow[n\\rightarrow \\infty]{} 0\n\\end{align*}\nPutting this together with (\\ref{PS2}), we get\n\\begin{align}\\label{PS3}\n\\frac{1}{2}\\int_M\\langle B,\\pounds_{X}g\\rangle\\mu_g=\\int_{\\partial M}B(X,\\nu)\\mu_{\\partial M}.\n\\end{align}\n\n\nNow, if we write $B=\\overset{\\circ}{B}+\\frac{1}{n}\\mathrm{tr}_gB \\: g$, where $\\overset{\\circ}{B}$ is the traceless part of $B$, we get the following\n\\begin{align}\\label{PS4}\n\\langle B, \\pounds_Xg \\rangle &= \\langle \\overset{\\circ}{B},\\pounds_X g \\rangle + \\frac{2}{n}\\mathrm{tr}_gB\\mathrm{div}_gX.\n\\end{align}\nNotice that $\\mathrm{tr}_gB\\mathrm{div}_gX=\\mathrm{div}_g(\\mathrm{tr}_gB X) - X(\\mathrm{tr}_gB)$, which gives us that \n\\begin{align*}\n\\int_M\\mathrm{tr}_gB\\mathrm{div}_gX\\mu_g=\\int_{\\partial M}\\mathrm{tr}_gB\\langle X,\\nu\\rangle \\mu_{\\partial M} -\\int_M X(\\mathrm{tr}_gB)\\mu_g \\;\\; \\forall \\;\\; B\\in C^{\\infty}_{0}.\n\\end{align*}\nDoing as above and approximating $B$ by compactly supported fields converging to $B$ in $H_{s+1,\\rho}$, we get\n\\begin{align*}\n\\Big\\vert\\int_M(\\mathrm{tr}_gB\\mathrm{div}_gX-\\mathrm{tr}_gB_n\\mathrm{div}_gX)\\mu_g\\Big\\vert&\\leq \\int_M|\\mathrm{tr}_g(B-B_n)||\\mathrm{div}_gX| \\mu_g,\\\\\n&\\lesssim \\int_M|B-B_n|_g|\\nabla X|_g\\mu_g,\\\\\n&\\leq ||B-B_n||_{L^2}||\\nabla X||_{L^2}\\xrightarrow[n\\rightarrow\\infty]{} 0,\n\\end{align*}\nsince, by hypothesis, $\\nabla X\\in L^2(M)$. The above shows that \n\\begin{align}\n\\int_M\\mathrm{tr}_gB\\mathrm{div}_gX\\mu_g=\\lim_{n\\rightarrow\\infty}\\Big\\{ \\int_{\\partial M}\\mathrm{tr}_gB_n\\langle X,\\nu\\rangle \\mu_{\\partial M} -\\int_M X(\\mathrm{tr}_gB_n)\\mu_g \\Big\\}.\n\\end{align}\nFurthermore, notice that \n\\begin{align*}\n\\Big\\vert\\int_M X(\\mathrm{tr}_g(B-B_n))\\mu_g\\Big\\vert&\\leq \\int_M|\\langle \\nabla\\mathrm{tr}_g(B-B_n),X\\rangle|\\mu_g,\\\\\n&\\lesssim \\int_M|\\nabla(B-B_n)|_g|X|_g\\mu_g,\\\\\n&=\\int_M|\\nabla(B-B_n)|_g(1+d^2_e)^{\\frac{1}{2}(\\rho+1)}(1+d^2_e)^{-\\frac{1}{2}(\\rho+1)}|X|_g\\mu_g,,\\\\\n&\\leq ||X||_{L^2_{-(\\rho+1)}}||\\nabla(B-B_n)||_{L^2_{\\rho+1}}\\xrightarrow[n\\rightarrow\\infty]{} 0\n\\end{align*}\n\nSimilarly, we also get that\n\\begin{align*}\n\\Big\\vert\\int_{\\partial M}\\mathrm{tr}_g(B - B_n)\\langle X,\\nu\\rangle \\mu_{\\partial M}\\Big\\vert&\\lesssim \\int_{\\partial M}|B - B_n|_g |X|_g,\\\\\n&\\leq ||B-B_n||_{L^2(\\partial M)} ||X||_{{L^2(\\partial M)}} \\xrightarrow[n\\rightarrow\\infty]{} 0,\n\\end{align*}\nwhere we have again used the continuity of the trace map. All of the above, gives us that\n\\begin{align}\n\\int_M\\mathrm{tr}_gB\\mathrm{div}_gX\\mu_g=\\int_{\\partial M}\\mathrm{tr}_gB\\langle X,\\nu\\rangle \\mu_{\\partial M} -\\int_M X(\\mathrm{tr}_gB)\\mu_g.\n\\end{align}\nThus, integrating (\\ref{PS4}) and applying the above identity, we see that the following holds\n\\begin{align*}\n\\int_M\\langle B, \\pounds_Xg \\rangle \\mu_g = \\int_M\\langle \\overset{\\circ}{B},\\pounds_X g \\rangle\\mu_g + \\frac{2}{n}\\int_{\\partial M}\\mathrm{tr}_gB\\langle X,\\nu\\rangle \\mu_{\\partial M} - \\frac{2}{n} \\int_M X(\\mathrm{tr}_gB)\\mu_g.\n\\end{align*}\nPutting this together with (\\ref{PS3}), we get that the Pohozaev-Schone equality is valid in the present context:\n\\begin{align}\n\\int_M\\langle \\overset{\\circ}{B},\\pounds_X g \\rangle\\mu_g = \\frac{2}{n} \\int_M X(\\mathrm{tr}_gB)\\mu_g + 2 \\int_{\\partial M}\\overset{\\circ}{B}(X,\\nu)\\mu_{\\partial M}\n\\end{align}\n\n\\end{proof}\n\n\n\\section{Applications} \\label{applications}\n\nBefore going into the applications of the above identity, we would like to state and give an independent proof of the following rigidity statement linked with AE-manifolds. It should be stressed that similar statements, with slightly different hypotheses, can be found in several papers, such as \\cite{Lee-Parker},\\cite{Bartnik},\\cite{volcomp}.\n\n\\begin{thm}\\label{Ricciflatrigidity}\nLet $(M,e)$ be a, $n$-dimensional manifold euclidean at infinity, with $n\\geq 3$, and consider a $H_{s,\\delta}$-AE metric $g$, with $s> \\frac{n}{2}+2$ and $\\delta>-\\frac{n}{2}$. Then, if $g$ Ricci-flat, it holds that $(M,g)$ is isometric to $\\mathbb{R}^n$ with its standard flat metric. \n\\end{thm} \n\n\\begin{proof}\nThis proof will be carried out via a number of well-known results. First, notice that under our hypotheses on $g$, its Ricci tensor can be written, on any particular end, in harmonic coordinates giving us an expression of the form\n\\begin{align}\n{\\mathrm{Ric}_g}_{ij}=-g^{ab}\\partial_a\\partial_bg_{ij} + Q_{ij}(g,\\partial g),\n\\end{align}\nwhere $Q(g,\\partial g)$ is a quadratic polynomial in $\\partial g$. Thus, in these coordinates, the operator appearing in the right hand side of the above identity is an elliptic second order operator acting on the metric component functions written in such coordinates. Nevertheless, since the left-hand side is a tensor, we get that if $\\mathrm{Ric}_g\\in L^2_{\\tau}$, so is the left hand side on any given coordinate system. Thus, via elliptic theory we can improve the decay of the metric. This can be achieved by appealing to Theorem 5.1 in \\cite{CB1} plus an inductive argument of the type of Lemma 3.6 in Appendix II of \\cite{CB2}. In any case, the final result is that we can guarantee that $g-e\\in H_{2,\\tau-2}$, as long as $\\tau-2<\\frac{n}{2}-2$. In particular, if $g$ is Ricci-flat, then $g-e\\in H_{2,\\rho}\\cap C^2$ for any $\\rho<\\frac{n}{2}-2$. This implies that, near infinity, $g_{ij}=\\delta_{ij}+o_2(|x|^{-(\\rho+\\frac{n}{2})})$. Thus, we get that\n\\begin{align*}\ng_{ij}=\\delta_{ij}+o_2(|x|^{-\\rho'}), \\;\\; \\forall \\;\\; \\rho'\\frac{n}{2}$, is $g$-divergence-free. Furthermore, let $X\\in H_{s+2,-1}$ and $\\mathrm{tr}_gB=0$. Then, under these assumptions, any $B$-generalized soliton is $B$-flat.\n\\end{thm}\n\\begin{proof}\nUnder our working hypotheses, since $H_{s+2,-1}\\hookrightarrow C^1_{-1}$, we have both that $X\\in H_{1,-1}\\cap C^1$. Thus, we know that the identity (\\ref{PS2}) holds. Hence, if $\\mathrm{tr}_gB=0$, we have that\n\\begin{align*}\n\\int_M\\langle \\overset{\\circ}{B},\\pounds_X g \\rangle\\mu_g=0.\n\\end{align*}\nFurthermore, since $X$ satisfies (\\ref{Bgensol}), the following identities holds\n\\begin{align*}\n\\langle \\overset{\\circ}{B},\\pounds_X g \\rangle &= \\langle B,\\pounds_{X,conf} g \\rangle,\\\\\n&=- \\frac{1}{2}\\langle \\pounds_Xg,\\pounds_{X,conf} g \\rangle,\\\\\n&=- \\frac{1}{2}\\langle \\pounds_{X,conf}g,\\pounds_{X,conf} g \\rangle,\n\\end{align*}\nwhere in the first line we have introduced the conformal Killing Laplacian, defined by $\\pounds_{X,\\text{conf}}g\\doteq \\pounds_Xg-\\frac{2}{n}\\mathrm{div}_gX g$, which defines a traceless tensor; in the second line we used the soliton equation (\\ref{Bgensol}); and in the last line we used this traceless property of the conformal Killing Laplacian. With the above, we see that (\\ref{Bgensol}) implies that \n\\begin{align*}\n\\int_M|\\pounds_{X,conf}g|_g^2\\mu_g=0,\n\\end{align*}\nwhich is equivalent to $X\\in H_{s+1,-1}$ being a conformal Killing field of $g$. But it is known that there are no non-trivial Killing fields with this asymptotic behaviour \\cite{Chris}-\\cite{maxwell}, thus $X=0$, and thus $B$-flat.\n\\end{proof}\n\nFrom this theorem, we get the following corollaries.\n\n\\begin{coro}\nConsider an $n$-dimensional AE-Riemannian manifold $(M,g)$ without boundary, with $g-e\\in H_{s+3,\\delta}$, with $s>\\frac{n}{2}$ and $\\delta>-\\min\\{\\frac{n}{2},2\\}$ and $n\\geq 3$. Then, if $R_g=0$, any Ricci almost soliton $X\\in H_{s+2,\\delta+1}$ on $(M,g)$ is isometric to $(\\mathbb{R}^n,e)$, with $e$ the euclidean metric.\n\\end{coro}\n\\begin{proof}\nFirst, recall a Ricci almost soliton structure on a Riemannian manifold $(M,g)$ is given by a vector field $X$ satisfying the equation\n\\begin{align}\\label{Riccisol1}\nRic_g+\\frac{1}{2}\\pounds_Xg=\\lambda g,\n\\end{align}\nfor some function $\\lambda$. Notice that the above equation is equivalent to the $G$-generalized soliton equation given by\n\\begin{align}\\label{Riccisol2}\nG_g +\\frac{1}{2}\\pounds_Xg=(\\lambda - \\frac{1}{2}R_g)g.\n\\end{align}\nwhere $G_g\\doteq \\mathrm{Ric}_g-\\frac{1}{2}R_gg$ stands for the Einstein tensor associated to the metric $g$. Notice that under our hypotheses for $g$, we have that $G_g\\in H_{s+1,0}$. Thus, from this fact and the hypotheses on $X$, we are under the hypotheses of Theorem \\ref{Bgensol}. Hence, if $\\mathrm{tr}_gG_g=\\frac{2-n}{2}R_g=0$, we have that $Ric_g=0$ as a consequence of the Theorem. Thus, from Theorem \\ref{Ricciflatrigidity}, we get that $(M,g)$ is isometric to $(\\mathbb{R}^n,e)$.\n\n\n\\end{proof}\n\nSimilarly, we can consider \\textit{Codazzi-generalized solitons} for some AE-hypersurface $(M^n,g)$ isometrically immersed in a Ricci-flat space $(\\tilde{M}^{n+1},\\tilde{g})$. In such a case, from Codazzi's equation, we know that the tensor $\\pi\\doteq K - \\mathrm{tr}_gK g$ satisfies the conservation equation\n\\begin{align}\\label{piconservation}\n\\mathrm{div}_g\\pi=0,\n\\end{align}\nwhere $K$ stands for the \\textit{extrinsic curvature} of the hypersurface $M\\hookrightarrow \\tilde{M}$, which is defined as $K\\doteq \\langle \\RN{2},n \\rangle$, where $\\RN{2}$ stands for the second fundamental form associated to the immersion, and $n$ for the unit normal field. Now, we define a Codazzi-generalized soliton structure on such an immersion as the choice of a vector field $X$ on $M$ satisfying the following soliton-type equation\n\\begin{align}\\label{codsol}\nK + \\frac{1}{2}\\pounds_Xg = \\lambda g,\n\\end{align} \nfor some function $\\lambda$ on $M$. \n\n\\begin{coro}\nConsider the setting described above, suppose that $M$ does not have a boundary, and suppose that $(M,g,X)$ provides a Codazzi-generalized soliton structure in a Ricci-flat space. Furthermore, suppose that $K\\in H_{s+1,0}$ and $X\\in H_{s+2,-1}$. Then, if $M$ is a minimal or maximal hypersurface\\footnote{We are enabling $\\tilde{g}$ to be either Riemannian or Lorentzian.}, that is, $\\mathrm{tr}_gK=0$, then $(M,g)$ is a totally geodesic scalar-flat hypersurface. In particular, if the ambient space is actually flat, then $(M,g)$ is isometric to $(\\mathbb{R}^n,e)$, with $e$ the euclidean metric.\n\\end{coro}\n\\begin{proof}\nSince our ambient manifold is Ricci-flat, we know that the Codazzi equation for hypersurfaces implies that (\\ref{piconservation}) holds. Furthermore, the hypothesis of $\\mathrm{tr}_gK=0$ implies that $\\pi=K$, and thus $K$ is a \\textit{conserved} symmetric tensor, which satisfies all the hypotheses demanded for the tensor field $B$ in Theorem \\ref{thm1}. Furthermore, following the same arguments as in the previous corollary, we have that $X$ also satisfies all the hypotheses imposed in Theorem \\ref{thm1}. Thus, using Theorem \\ref{Bgensol}, we immediately get that $M\\hookrightarrow\\tilde{M}$ is totally geodesic. Now, notice that the Gauss equation can be written en the following way. Given $X,Y,Z,W\\in\\Gamma(TM)$, we have that\n\\begin{align}\\label{Gauss}\n\\tilde{g}(\\tilde{R}(X,Y)Z,W)=g(R(X,Y)Z,W) + \\epsilon\\big( K(X,Z)K(Y,W) - K(Y,Z)K(X,W) \\big),\n\\end{align}\nwhere $\\epsilon=\\tilde{g}(n,n)=\\pm 1$, depending on whether the normal direction is \\textit{space-like} or \\textit{time-like}. Now, we have just shown that $M$ is totally geodesic, thus $K=0$. Thus, consider an orthonormal frame $\\{E_{\\alpha}\\}_{\\alpha=0}^n$, where $E_{0}=n$ and thus $E_i$ is tangent to $M$ $\\forall$ $i=1,\\cdots,n$, so that we can write\n\\begin{align*}\n\\mathrm{Ric}_g(X,Y)&=\\sum_{i=1}^n\\tilde{g}(\\tilde{R}(E_i,Y)Z,E_i),\\\\\n&=\\mathrm{Ric}_{\\tilde{g}}(X,Y)-\\epsilon\\tilde{g}(\\tilde{R}(n,Y)Z,n),\\\\\n&=-\\epsilon\\tilde{g}(\\tilde{R}(n,Y)Z,n),\n\\end{align*} \nwhere in the third line we have used the hypotheses that $\\tilde{g}$ is Ricci-flat. From this, we get\n\\begin{align*}\nR_g=-\\epsilon\\sum_{j=1}^n\\tilde{g}(\\tilde{R}(n,E_j)E_j,n)=-\\epsilon\\sum_{j=1}^n\\tilde{g}(\\tilde{R}(E_j,n)n,E_j)=-\\epsilon\\mathrm{Ric}_{\\tilde{g}}(n,n)=0,\n\\end{align*}\nwhich proves that $(M,g)$ is scalar-flat. Finally, since $K=0$, it is a trivial consequence of (\\ref{Gauss}) that if $\\mathrm{Riem}_{\\tilde{g}}=0$, then $\\mathrm{Riem}_{g}=0$ and thus $(M^n,g)$ is isometric to $(\\mathbb{R}^n,e)$.\n\n\\end{proof}\n\\begin{remark}\nIn order to gain some insight about the kind of geometric picture that a Codazzi-soliton represents, it is an interesting observation to note that, following closely Al\\'ias-Lira-Rigoli \\cite{ALR}, we can understand (\\ref{codsol}) following the kind of definition provided in \\cite{ALR} for mean curvature flow solitons. That is, if, given an immersion $\\psi:M^n\\mapsto \\tilde{M}^{n+1}$, we say it defines a Codazzi-soliton with respect to a conformal Killing field $X\\in\\Gamma(T\\tilde{M})$ if it satisfies\n\\begin{align*}\nX^{\\perp}=-n,\n\\end{align*}\nalong $\\psi$, where $X^{\\perp}$ denotes the orthogonal component of $X$ to $\\psi(M)$ and $n$ denotes the unit normal of $\\psi(M)$. Then, following the same line of argument presented in \\cite{ALR}, equation (\\ref{codsol}) is an equation that any Codazzi-soliton must satisfy. Thus, in this sense, Codazzi solitons stand in perfect analogy with interesting soliton-like equations related to interesting geometric flows. \n\n\\end{remark}\n\n\\begin{remark}\nTaking into account the above remark and corollary, it is also interesting to note that, in the case of manifolds euclidean at infinity, admitting a scalar-flat metric is not a trivial statement. This can be viewed as a consequence of some important results, concerning for instance the Yamabe classification for AE manifolds \\cite{Maxwell-Dilts} and topological obstructions related with the positive mass theorem \\cite{Schoen-Yau}. More explicitly, using the Yamabe classification provided in \\cite{Maxwell-Dilts}, we know that AE manifolds produced by removing points from closed manifolds which do not admit Yamabe positive metric do not carry any metric of zero scalar curvature. Using the results established in \\cite{Schoen-Yau}, it is possible to produce a wide variety of closed manifolds which do not admit a Yamabe positive metric. For instance, their main claim is that manifolds of the form $M^n\\# T^n$, with $M^n$ closed, lie in this category. Now, all this implies that AE manifolds obtained by removal of a finite number of points from such closed manifolds do not admit any Codazzi-soliton structure in any Ricci-flat space. \n\\end{remark}\n\nSimilarly to what we have done above, we could easily define new soliton-type equation related with other conserved second rank tensor fields, which we could relate to other generalized geometric flows. \n\n\\subsection*{Generalized Almost-Schur type inequality}\n\nThe classical Schur's lemma says that every connected Einstein manifold of dimension $n\\geq 3$ has constant scalar curvature. In \\cite{Topping}, De Lellis and Topping studied a quantitative version of the classical Schur Lemma, that allows to infer that, if a closed Riemannian manifold is close to being Einstein in the $L^{2}$ sense, then its scalar curvature is close to a constant in $L^{2}$. In this scope, they proved the so called ``Almost Schur Lemma\" that claims if $(M^{n},g)$ is a closed Riemannian manifold of dimension $n\\geq 3$, with nonnegative Ricci curvature, then\n\n$$\\displaystyle\\int_{M}(R-\\bar{R})^{2}\\mu_{g}\\leq \\frac{4n(n-1)}{(n-2)^{2}}\\displaystyle\\int_{M}\\left|Ric-\\frac{R}{n}g\\right|^{2}\\mu_{g},$$\nwhere we denote $\\bar{R}=\\frac{1}{Vol M}\\int_{M}R\\mu_{g}$. Furthermore, the equality occurs if, and only if, $M$ is an Einstein manifold.\n\nWe observe that a central step in the proof is the following integral identity (see \\cite{Topping})\n\n$$-\\displaystyle\\int_{M}\\langle dR, df\\rangle \\mu_{g}=\\frac{2n}{n-2}\\displaystyle\\int_{M}\\langle{\\stackrel{\\circ}{Ric}},D^{2}f\\rangle,$$\nwhere $f$ is a solution of a PDE. In fact, this integral formula is the Pohoz\\v{a}ev-Sch\\\"oen identity, if $X=\\nabla f$ and $\\partial M=\\emptyset$. In fact, this idea can be used to obtain these type of inequalities for more general tensors and under others suitable conditions on the Ricci curvature (\\cite{Barbosa},\\cite{Cheng1},\\cite{Cheng2}). The idea of this section is to generalize these type of inequalities obtained in \\cite{Topping} to non-compact manifolds in our case of interest. Thus, suppose that $(M^n,g)$ is an $H_{s+3,\\delta}$-asymptotically euclidean manifold, with $n\\geq 3$, $\\delta>-\\frac{n}{2}$, $\\delta\\geq -2$ and $s>\\frac{n}{2}$. Recall from \\cite{CB1} that $\\Delta_g:H_{s+2,\\delta_0}\\mapsto H_{s,\\delta_0+2}$ is an isomorphism for $-\\frac{n}{2}<\\delta_0<\\frac{n}{2}-2$. It will be an interesting observation to note that this is not true for closed manifolds, where what we have is that the operators of the form $\\Delta_g-a:H_{s+2}\\mapsto H_s$ with $a>0$ and $a\\in H_{s}$ have this isomorphism property. This difference can be traced back to the fact that on closed manifolds constant functions are in the Kernel of $\\Delta_g$, while on AE-manifolds we have a Poincar\\'e inequality \\cite{Bartnik}-\\cite{Maxwell-Dilts}, which guarantees that $\\Delta_g$ has the isomorphism property. In fact, a stronger statement that can be made. That is, given a $H_{s+1,\\delta}$-AE metric, with $s>\\frac{n}{2}$ and $\\delta>-\\frac{n}{2}$, then the Laplacian operator associated the $g$, that is $\\Delta_g$, is injective on $H_{2,\\rho}$ for any $\\rho>-\\frac{n}{2}$. This result is a consequence of Corallary 4.3 presented in Appendix 2 of \\cite{CB2} and an application of the Sobolev embedding theorems. These points are key features that will allow us to produce an almost-Schur-type inequality without any restrictions on the Ricci-curvature. It is known that, for $n\\geq 5$, being bounded from below is a necessary condition in the case of closed manifolds \\cite{Topping}. \n\n\\begin{thm}\\label{B-AS}\nLet $(M^n,g)$ is an $H_{s+3,\\delta}$-asymptotically euclidean manifold without boundary, with $n\\geq 3$, $s>\\frac{n}{2}$, $\\delta>-\\frac{n}{2}$ and $\\delta\\geq -2$. Now, let $B$ be a symmetric $(0,2)$-tensor field satisfying the hypotheses of Theorem \\ref{thm1}, that is, $\\mathrm{div}_gB=0$ and $B\\in H_{s+1,\\rho}$, with $\\rho\\geq 0$, furthermore, suppose that $\\rho>-\\frac{n}{2} + 2$, and denote its trace by $b\\doteq \\mathrm{tr}_gB$. Then:\\\\\nI) there is a constant $C$ independent of the specific choice of such tensor $B$, such that the following inequality holds\n\\begin{align}\\label{AS}\n||b||_{L^2}\\leq n\\left( \\frac{n-1}{n} + C||\\mathrm{Ric}_g||_{C^0_2}\\right)^{\\frac{1}{2}} ||\\overset{\\circ}{B}||_{L^2},\n\\end{align}\nII) If $\\mathrm{Ric}_g$ is non-negative, \\textit{i.e}, $\\mathrm{Ric}_g(X,X)\\geq 0$ $\\forall$ $X\\in\\Gamma(TM)$, then\n\\begin{align}\\label{AS.01}\n||b||_{L^2}\\leq \\sqrt{n(n-1)} ||\\overset{\\circ}{B}||_{L^2}.\n\\end{align}\n\\end{thm}\n\n\\begin{proof}\nUnder the above hypotheses, since $\\delta>-\\frac{n}{2}$, we have that $b\\in H_{s+1,\\rho}$. Now, assume that $\\rho<\\frac{n}{2}$. If actually $\\rho\\geq \\frac{n}{2}$, then we would have a continuous embedding $H_{s+1,\\rho}\\hookrightarrow H_{s+1,\\sigma}$, for any $\\sigma<\\frac{n}{2}$. Thus, this assumption posses no actual restrictions, and from now on, so as to avoid introducing one further weighting parameter, we will assume it. Also, from the hypotheses bounding $\\rho$ from below, we have that $-\\frac{n}{2}+2<\\rho<\\frac{n}{2}$, for $n=3,4$; and $0\\leq\\rho<\\frac{n}{2}$, for $n\\geq 5$. Notice that these conditions can always be satisfied, and that, as stated before, we are not posing any further restrictions. \n\nNow, consider $\\Delta_g:H_{s+3,\\rho-2}\\mapsto H_{s+1,\\rho}$, and consider the equation\n\\begin{align}\\label{AS1}\n\\Delta_gf=b.\n\\end{align}\nSince $-\\frac{n}{2}<\\rho-2<\\frac{n}{2}-2$ for any $n\\geq 3$, then, from the discussion preceding the theorem, we know that there is a solution $f\\in H_{s+3,\\rho-2}$. Hence, we get the following:\n\\begin{align}\\label{AS2}\n\\begin{split}\n\\int_M b^2\\mu_g&=\\int_Mb\\Delta_gf\\mu_g,\\\\\n&=-\\int_M\\nabla f(b)\\mu_g + \\int_M\\mathrm{div}_g(b\\nabla f)\\mu_g.\n\\end{split}\n\\end{align} \nNow, to see that the last term actually does not produce any contribution, the procedure is standard: Consider sequences $\\{b_n\\}\\subset C^{\\infty}_0$ of compactly supported functions converging to $b$ in $H_{s+1,\\rho}$ and $\\{\\nabla f_n\\}\\subset C^{\\infty}_0$ converging to $\\nabla f$ in $H_{s+2,\\rho-1}$. Then, consider the difference\n\\begin{align}\\label{AS3}\n\\begin{split}\n\\Big\\vert\\int_M\\{\\mathrm{div}_g(b\\nabla f)-\\mathrm{div}_g(b_n\\nabla f_n)\\}\\mu_g\\Big\\vert\\leq &\\int_M|\\mathrm{div}_g(b(\\nabla f-\\nabla f_n)|\\mu_g \\\\\n&+ \\int_M|\\mathrm{div}_g((b-b_n)\\nabla f_n)|\\mu_g.\n\\end{split}\n\\end{align}\nWe also have that $\\mathrm{div}_g(b(\\nabla f-\\nabla f_n))=\\langle\\nabla b,\\nabla f-\\nabla f_n\\rangle + b\\:\\mathrm{div}_g(\\nabla f-\\nabla f_n)$, which gives point wise estimates of the form:\n\\begin{align*}\n|\\mathrm{div}_g(b(\\nabla f-\\nabla f_n))|\\lesssim |\\nabla f-\\nabla f_n|_g|\\nabla b|_g + |b||\\nabla(\\nabla f-\\nabla f_n)|_g.\n\\end{align*}\nNotice that, since $\\rho\\geq 0$, then $H_{s+3,\\rho-1}\\hookrightarrow L^2_{-1}$ and $H_{s,\\rho+1}\\hookrightarrow L^2_{1}$. Thus, after integration, we get the following estimate:\n\\begin{align*}\n\\int_M|\\mathrm{div}_g(b(\\nabla f-\\nabla f_n)|\\mu_g&\\lesssim ||\\nabla f-\\nabla f_n||_{L^2_{-1}}||\\nabla b||_{L^2_1} \\\\\n&+ ||b||_{L^2}|||\\nabla(\\nabla f - \\nabla f_n)||_{L^2}\\xrightarrow[n\\rightarrow\\infty]{} 0.\n\\end{align*}\nA similar argument can be applied to the second term in the right-hand side of (\\ref{AS3}), showing that\n\\begin{align*}\n\\int_M\\mathrm{div}_g(b\\nabla f)\\mu_g = \\lim_{n\\rightarrow\\infty} \\int_M\\mathrm{div}_g(b_n\\nabla f_n)\\mu_g=0.\n\\end{align*} \nThus, going back to (\\ref{AS2}), we get the following:\n\\begin{align}\\label{AS4}\n\\begin{split}\n\\int_M b^2\\mu_g&=-\\int_M\\nabla f(b)\\mu_g,\\\\\n&=-\\frac{n}{2}\\int_M\\langle \\overset{\\circ}{B},\\pounds_{\\nabla f}g \\rangle\\mu_g,\\\\\n&=-n\\int_M\\langle \\overset{\\circ}{B},\\nabla^2f \\rangle\\mu_g,\\\\\n&=-n\\int_M\\langle \\overset{\\circ}{B},\\nabla^2f - \\frac{1}{n}g\\Delta_gf \\rangle\\mu_g\n\\end{split}\n\\end{align}\nwhere in the second line we have used (\\ref{PS}). Clearly, since the left hand side is non-negative, then we have that\n\\begin{align}\\label{AS5}\n\\begin{split}\n0\\leq -n\\int_M\\langle \\overset{\\circ}{B},\\nabla^2f - \\frac{1}{n}g\\Delta_gf \\rangle\\mu_g &= n\\Big\\vert\\int_M\\langle \\overset{\\circ}{B},\\nabla^2f - \\frac{1}{n}g\\Delta_gf \\rangle\\mu_g\\Big\\vert,\\\\\n&\\leq n\\int_M |\\overset{\\circ}{B}|_g |\\nabla^2f - \\frac{1}{n}g\\Delta_gf|_g \\mu_g,\\\\\n&\\leq n||\\overset{\\circ}{B}||_{L^2}||\\nabla^2f - \\frac{1}{n}g\\Delta_gf||_{L^2}.\n\\end{split}\n\\end{align}\nNotice that due to the choice of our functional spaces the right-hand side is well-defined, since both $|\\overset{\\circ}{B}|_g,|\\nabla^2f|_g\\in L^2$. Also, since $H_{s+3,\\rho-2}\\subset C^{3}$, we get the usual Ricci identity\n\\begin{align*}\n{\\mathrm{Ric}_g}_{uj}\\nabla^uf=\\nabla_i\\nabla_j\\nabla^if - \\nabla_j(\\Delta_gf),\n\\end{align*}\nwhich implies \n\\begin{align}\\label{ricciid}\n{\\mathrm{Ric}_g}(\\nabla f,\\nabla f)=\\langle \\nabla f,\\mathrm{div}_g\\left(\\nabla^2 f\\right)\\rangle_g - \\langle \\nabla f,\\nabla(\\Delta_gf)\\rangle_g.\n\\end{align}\nSince $\\nabla f\\in H_{s+2,\\rho-1}$ and $\\mathrm{Ric}_g\\in H_{s+1,\\delta+2}$, with $\\delta> -\\frac{n}{2}$ and $\\rho\\geq 0$, then ${\\mathrm{Ric}_g}(\\nabla f,\\nabla f)\\in H_{s+1,\\sigma}$ for any $\\sigma< \\delta + \\rho + \\frac{n}{2}$, which implies ${\\mathrm{Ric}_g}(\\nabla f,\\nabla f)\\in L^2$. Furthermore, since $\\nabla^2f\\in H_{s+1,\\rho}$, then both $\\nabla^3f$ and $\\nabla(\\Delta_gf)$ lie in $H_{s,\\rho+1}\\hookrightarrow L^{2}_1$. Thus, since $\\nabla f\\in L^2_{-1}$, then, both terms in the right hand side of (\\ref{ricciid}) are actually in $L^1$, since \n\\begin{align*}\n\\Big\\vert \\int_M\\langle \\nabla f,\\mathrm{div}_g\\nabla^2 f\\rangle_g\\mu_g\\Big\\vert&\\leq \\int_M |\\nabla f| |\\mathrm{div}_g\\nabla^2 f| \\mu_g \\lesssim ||\\nabla f||_{L^2_{-1}}||\\nabla^3f||_{L^2_{1}}<\\infty,\\\\\n\\Big\\vert \\int_M\\langle \\nabla f,\\nabla(\\Delta_gf)\\rangle_g\\mu_g\\Big\\vert&\\lesssim ||\\nabla f||_{L^2_{-1}}||\\nabla (\\Delta_gf)||_{L^2_{1}}<\\infty.\n\\end{align*}\nThis implies that we can integrate (\\ref{ricciid}) over $M$. Then, we can integrate by parts, so as to get\n\\begin{align*}\n\\int_M\\langle \\nabla f,\\mathrm{div}_g\\nabla^2 f\\rangle_g\\mu_g= - \\int_M|\\nabla^2f|^2_g\\mu_g + \\int_M\\mathrm{div}_g\\left(i_{\\nabla f}\\nabla^2f \\right)\\mu_g.\n\\end{align*}\nNotice that from the above estimates, approximating $f$ in $H_{s+3,\\rho-2}$ by $\\{f_n \\}_{n=1}^{\\infty}\\subset C^{\\infty}_0$, we get that\n\\begin{align*}\n\\Big\\vert \\int_M\\left(\\langle \\nabla f,\\mathrm{div}_g\\nabla^2 f \\rangle_g - \\langle \\nabla f_n,\\mathrm{div}_g\\nabla^2 f_n\\rangle_g\\right)\\mu_g\\Big\\vert&\\lesssim ||\\nabla (f-f_n)||_{L^2_{-1}}||\\nabla^3f||_{L^2_{1}} \\\\\n&+ ||\\nabla f_n||_{L^2_{-1}}||\\nabla^3(f-f_n)||_{L^2_{1}}\\xrightarrow[n\\rightarrow\\infty]{}0,\n\\end{align*}\nwhich implies that\n\\begin{align*}\n\\int_M\\langle \\nabla f,\\mathrm{div}_g\\nabla^2 f \\rangle_g\\mu_g=-\\lim_{n\\rightarrow\\infty}\\int_M|\\nabla^2f_n|^2_g\\mu_g=-\\int_M|\\nabla^2f|^2_g\\mu_g,\n\\end{align*}\nwhere the last equality comes from the same kind of approximation argument, plus the fact that $\\nabla^2f\\in L^2$. Following the lines as described above, we also get that\n\\begin{align*}\n\\int_M\\langle \\nabla f,\\nabla(\\Delta_gf)\\rangle_g\\mu_g=-\\lim_{n\\rightarrow\\infty}\\int_{M}|\\Delta_gf_n|^2\\mu_g=-\\int_{M}|\\Delta_gf|^2\\mu_g.\n\\end{align*}\nThus, finally, we get that the following expression, familiar for compact manifolds, holds under our hypotheses\n\\begin{align}\n\\int_M{\\mathrm{Ric}_g}(\\nabla f,\\nabla f)\\mu_g= \\int_{M}|\\Delta_gf|^2\\mu_g - \\int_M|\\nabla^2f|^2_g\\mu_g.\n\\end{align}\nNoticing that $|\\nabla^2f-\\frac{1}{n}g\\Delta_gf|^2_g=|\\nabla^2f|^2_g - \\frac{1}{n} |\\Delta_gf|^2$, we get\n\\begin{align*}\n||\\nabla^2f-\\frac{1}{n}g\\Delta_gf||^2_{L^2}&=||\\nabla^2f||^2_{L^2} - \\frac{1}{n} ||\\Delta_gf||^2_{L^2}\\\\\n&=\\frac{n-1}{n} ||\\Delta_gf||^2_{L^2} - \\int_M{\\mathrm{Ric}_g}(\\nabla f,\\nabla f)\\mu_g.\n\\end{align*}\nThus, notice that, parallel to what happens in the compact scenario, if $\\mathrm{Ric}_g$ is non-negative, then we can get an estimate of the $L^2$-norm of $\\nabla^2f-\\frac{1}{n}g\\Delta_gf$ in terms of the $L^2$ norm of $\\Delta_gf$. Doing this, we can prove the second statement in the Lemma. That is, if $\\mathrm{Ric}_g$ is non-negative, then putting the above identity together with (\\ref{AS4})-(\\ref{AS5}), we get\n\\begin{align}\n||b||^2_{L^2}\\leq \\sqrt{n(n-1)}||\\overset{\\circ}B||_{L^2}||b||_{L^2}.\n\\end{align}\n\nNevertheless, we can actually get rid of this limitation on the Ricci tensor with the following argument. Notice that, from the point wise continuity of contractions, we get\n\\begin{align*}\n\\Big\\vert \\int_M\\mathrm{Ric}_g(\\nabla f,\\nabla f)\\mu_g \\Big\\vert \\lesssim \\int_{M}|\\mathrm{Ric}_g|_g|\\nabla f|^2_g\\mu_g. \n\\end{align*}\nAlso, since $\\mathrm{Ric}_g\\in H_{s+1,\\delta+2}$, with $s>\\frac{n}{2}$ and $\\delta>-\\frac{n}{2}$, then $H_{s+1,\\delta+2}\\hookrightarrow C^0_{2}$, where \n\\begin{align*}\n||\\mathrm{Ric}_g||_{C^0_2}=\\sup_{M}|\\mathrm{Ric}_g|_g(1+d^2_e),\n\\end{align*}\nthus, we get\n\\begin{align*}\n\\Big\\vert \\int_M\\mathrm{Ric}_g(\\nabla f,\\nabla f)\\mu_g \\Big\\vert &\\lesssim ||\\mathrm{Ric}_g||_{C^0_2}\\int_M|\\nabla f|^2_g(1+d^2)^{-1}\\mu_g\\\\\n&=||\\mathrm{Ric}_g||_{C^0_2}\\int_M|\\nabla f|^2_g(1+d^2)^{\\rho-1}(1+d^2)^{-\\rho}\\mu_g\\\\\n&\\leq ||\\mathrm{Ric}_g||_{C^0_2}||\\nabla f||^2_{L^2_{\\rho-1}}.\n\\end{align*}\nNotice, again, that since $f\\in H_{s+3,\\rho-2}$, with $\\rho\\geq 0$, then $\\nabla f\\in L^2_{\\rho-1}$. Thus, the above expression is well-defined. We then get the following estimate, where $C_n$ stands for some constant depending only on $n$,\n\\begin{align*}\n||\\nabla^2f-\\frac{1}{n}g\\Delta_gf||^2_{L^2}&\\leq \\frac{n-1}{n} ||\\Delta_gf||^2_{L^2} + C_{n}||\\mathrm{Ric}_g||_{C^0_2}||\\nabla f||^2_{L^2_{\\rho-1}},\\\\\n&\\leq \\frac{n-1}{n} ||\\Delta_gf||^2_{L^2} + C_n||\\mathrm{Ric}_g||_{C^0_2}||f||^2_{H_{2,\\rho-2}}\n\\end{align*}\nNow, notice that $\\rho-2>-\\frac{n}{2}$, and since $\\Delta_g:H_{2,\\rho-2}\\mapsto H_{0,\\rho}$ is injective for $\\rho-2>-\\frac{n}{2}$, we have the following elliptic estimate (see \\cite{CB1}):\n\\begin{align*}\n||f||_{H_{2,\\rho-2}}\\leq C ||\\Delta_gf||_{H_{0,\\rho}}\\leq C ||b||_{L^2},\n\\end{align*}\nfor some constant $C$, which does not depend on the particular $f\\in H_{2,\\rho-2}$, where the last inequality is a consequence of the embedding $L^2_{\\rho}\\subset L^2$ for $\\rho\\geq 0$. This implies that\n\\begin{align*}\n||\\nabla^2f-\\frac{1}{n}g\\Delta_gf||_{L^2}&\\leq \\left( \\frac{n-1}{n} + C||\\mathrm{Ric}_g||_{C^0_2}\\right)^{\\frac{1}{2}}||b||_{L^2},\n\\end{align*}\nFor some other constant $C$. Putting this together with (\\ref{AS4})-(\\ref{AS5}), we get\n\\begin{align*}\n||b||^2_{L^2}\\leq n\\left( \\frac{n-1}{n} + C||\\mathrm{Ric}_g||_{C^0_2}\\right)^{\\frac{1}{2}}||\\overset{\\circ}{B}||_{L^2}||b||_{L^2} , \n\\end{align*}\n\\end{proof}\n\n\\begin{coro}\nUnder the same hypotheses of the previous lemma, if $n\\geq 5$, then (\\ref{AS.01}) can be simplified by the following estimate\n\\begin{align}\n||b||^2_{L^2}\\leq nC_g ||\\overset{\\circ}B||_{L^2},\n\\end{align}\nwhere the constant $C_g$ is the best constant satisfying the elliptic estimate $||h||_{H_{2,-2}}\\leq C_g||\\Delta_gh||$ for all $h\\in H_{2,-2}$.\n\\end{coro}\n\\begin{proof}\nUnder these assumptions we can look at $b\\in H_{s+1,0}$, and consider the map $\\Delta_g:H_{s+3,-2}\\mapsto H_{s+1,0}$, which will give us a solution $f\\in H_{s+3,-2}$ to (\\ref{AS1}). Then, from (\\ref{AS4})-(\\ref{AS5}), we can actually deduce that\n\\begin{align*}\n||b||^2_{L^2}\\leq n ||\\overset{\\circ}B||_{L^2}||\\nabla^2f||_{L^2}.\n\\end{align*}\nThus, since $||\\nabla^2f||_{L^2}\\leq ||\\nabla^2f||_{H_{2,-2}}$, and, again, for $n\\geq 5$ we have that $\\Delta_g$ is injective on $H_{2,-2}$, we get the elliptic estimate $||\\nabla^2f||_{H_{2,-2}}\\leq C_g||b||_{L^2}$. In this case, we would directly get\n\\begin{align}\n||b||^2_{L^2}\\leq nC_g ||\\overset{\\circ}B||_{L^2}||b||_{L^2},\n\\end{align}\nwhich is a simpler estimate than the one presented in the main proof.\n\\end{proof}\n\nFrom this general theorem, we get a version of the usual almost-Schur inequality for the scalar-curvature \\cite{Topping}-\\cite{Barbosa} in the context of AE manifolds. In our case, we are able to state such inequality without restriction on the Ricci tensor, but we do not provide an explicit value for the constant $C$. \n\n\\begin{coro}\\label{R-AS}\nLet $(M^n,g_0)$ is an $H_{s+3,\\delta}$-asymptotically euclidean manifold without boundary, with $n\\geq 3$, $\\delta>-\\frac{n}{2}$ and $\\delta\\geq-2$ and $s>\\frac{n}{2}$. Then, there is a constant $C$, such that for any $H_{s+3,\\delta}$-AE metric $g$ on $M$ the following inequality holds\n\\begin{align}\n||R_g||_{L^2}\\leq C ||\\overset{\\circ}{\\mathrm{Ric}_g}||_{L^2}\n\\end{align}\n\\end{coro}\n\\begin{proof}\nNotice that since $g$ is $H_{s+3,\\delta}$-AE, with $\\delta>-\\frac{n}{2}$ and $\\delta\\geq -2$, then we have $\\mathrm{Ric}_g,R_g\\in H_{s+1,\\rho}$, for $\\rho>-\\frac{n}{2}+2$ and $\\rho\\geq 0$, which also implies $G_g\\in H_{s+1,\\rho}$ for the same $\\rho$. Hence, since $\\mathrm{div}_gG_g=0$, we get as an immediate corollary of the above Lemma that\n\\begin{align*}\n||R_g||_{L^2}\\leq C ||\\overset{\\circ}{\\mathrm{G}_g}||_{L^2}.\n\\end{align*}\nFinally, since $\\overset{\\circ}{\\mathrm{G}_g}=\\overset{\\circ}{\\mathrm{Ric}_g}$ our claim holds.\n\\end{proof}\n\nClearly, the following also holds as a direct consequence of the above Lemma.\n\\begin{coro}\nLet $(M^n,g)$ is an $H_{s+3,\\delta}$-asymptotically euclidean manifold without boundary, with $n\\geq 3$, $\\delta> - \\frac{n}{2}$, $\\delta\\geq -2$ and $s>\\frac{n}{2}$. Suppose that there is an isometric immersion $(M^n,g)\\hookrightarrow (\\tilde{M}^n,\\tilde{g})$ into a Ricci-flat space, \\textit{i.e}, $\\mathrm{Ric}_{\\tilde{g}}=0$. Then, if the extrinsic curvature $K\\in H_{s+1,\\rho}$ for $\\rho>-\\frac{n}{2}+2$ and $\\rho\\geq 0$, then, there is a constant $C$, independent of the immersion, such that\n\\begin{align}\\label{AS-K}\n||\\tau||_{L^2}\\leq C ||\\overset{\\circ}{K}||_{L^2},\n\\end{align} \nwhere $\\tau\\doteq\\mathrm{tr}_gK$.\n\\end{coro}\n\\begin{proof}\nAs we have already commented, under our hypotheses, we have that the Codazzi equation for hypersurfaces guarantees that the tensor $\\pi=K - \\tau g\\in H_{s+1,0}$ and satisfies $\\mathrm{div}_g\\pi=0$. Furthermore, we have that $\\overset{\\circ}{\\pi}=\\overset{\\circ}{K}$. Thus (\\ref{AS}) proves (\\ref{AS-K}).\n\\end{proof}\n\nFinally, we will present an almost-Schur-type inequality for $Q$-curvature on AE-manifolds. Following \\cite{Topping}-\\cite{Barbosa}, it has been shown in \\cite{chinos} that such an inequality holds in the context of closed manifolds. \nIt should be stress that problems related with $Q$-curvature have gained plenty of attention, see \\cite{Qcurv} for a survey. Before presenting this result, we should introduce some definitions. First of all, given a Riemannian manifold $(M^n,g)$, with $n\\geq 3$, we will define its $Q$-curvature in the following way\n\\begin{align}\\label{Qcurv}\nQ_g\\doteq A_n\\Delta_gR_g + B_n|\\mathrm{Ric}_g|^2_g + C_nR^2_g,\n\\end{align}\nwhere $A_n=-\\frac{1}{2(n-1)}$, $B_n=-\\frac{2}{(n-2)^2}$ and $C_n=\\frac{n^2(n-4)+16(n-1)}{8(n-1)^2(n-2)^2}$. \n\nNow, following \\cite{chinos}, we will explain how we can canonically associate to $Q$-curvature a divergence-free $(0,2)$-symmetric tensor field.\nNotice that we can view $Q$-curvature as a map on the set of Riemannian metrics on a given manifold $M$, given by\n\\begin{align*}\nQ:\\mathrm{Riem}^{k}(M)&\\mapsto C^{k-4}(M),\\\\\ng &\\mapsto Q_g,\n\\end{align*} \nwhere $\\mathrm{Riem}^{k}(M)$ denotes the set of $C^k$ Riemannian metrics on $M$ and we are considering $k\\geq 4$. Then, we can analyse its linearisation $L_g\\doteq D_gQ:S^2M\\mapsto C^{k-4}(M)$ which is given by an operator acting on the space of $C^k$-symmetric tensor fields on $M$. Let $L^{*}_g$ be its formal adjoint operator, which acts on functions on $M$ and produces a symmetric tensor-field on $M$. Now, consider any point $p\\in M$ and a bounded neighbourhood $U$ of $p$. Furthermore, let $V\\subset U$ be another neighbourhood of $p$. Let $f\\in C^{\\infty}_{0}(U)$ and $X$ be a compactly supported smooth vector field on $U$. Then, we have that\n\\begin{align*}\n\\int_U\\langle X,\\mathrm{div}_g(L^{*}_gf) \\rangle \\mu_g = -\\frac{1}{2}\\int_U\\langle \\pounds_Xg,L^{*}_gf \\rangle \\mu_g=-\\frac{1}{2}\\int_U f L_g(\\pounds_Xg) \\mu_g.\n\\end{align*}\nNow, let $\\psi_t$ be the flow associated to the vector field $X$, then, we have that\n\\begin{align*}\nX(Q_g)=\\frac{d}{dt}\\psi^{*}_t(Q_{g})|_{t=0}=\\frac{d}{dt}Q_{\\psi^{*}_tg}|_{t=0}=D_{\\psi^{*}_tg}Q\\cdot\\frac{d}{dt}\\psi^{*}_tg|_{t=0}=D_{g}Q\\cdot\\pounds_Xg=L_g(\\pounds_Xg),\n\\end{align*}\nthus, we have that\n\\begin{align*}\n\\int_U\\langle \\mathrm{div}_g(L^{*}_gf),X \\rangle \\mu_g = -\\frac{1}{2}\\int_U \\langle fdQ_g, X \\rangle \\mu_g, \n\\end{align*}\nwhich implies that $\\mathrm{div}_g(L^{*}_gf)+\\frac{1}{2}fdQ_g=0$ $\\forall$ $f\\in C^\\infty_0(U)$. In particular, if we consider $f|_{V}\\equiv 1$, then we get that, in a neighbourhood of $p$, it holds that\n\\begin{align*}\n\\mathrm{div}_g(L^{*}_g1)+\\frac{1}{2}dQ_g=\\mathrm{div}_g\\big( L^{*}_g1 + \\frac{1}{2}Q_gg \\big) =0.\n\\end{align*}\nIn particular, this means that if we define the tensor field $J_g\\doteq -\\frac{1}{2}L^{*}_g1$ then, $B_J\\doteq J_g - \\frac{1}{4}Q_gg$ is a conserved $(0,2)$-symmetric tensor field. The explicit expression for such tensor field is given in terms of the $Q$-curvature, the Bach tensor and the Schoutten tensor, and, as a map on the metric, is a fourth order operator. Explicitly, it can be written as follows \\cite{chinos}.\n\\begin{align*}\n\\begin{split}\n&J_g=\\frac{1}{n}Q_gg-\\frac{1}{n-2}B_g-\\frac{n-4}{4(n-1)(n-2)}T_g,\\\\\n&{B_g}_{jk}\\doteq \\nabla^i{C_g}_{ijk} + {W_g}_{ijkl}S_g^{il},\\\\\n&{C_g}_{ijk}\\doteq \\nabla_i{S_g}_{jk} - \\nabla_j{S_g}_{ik},\\\\\n&S_g\\doteq \\frac{1}{n-2}(\\mathrm{Ric}_g - \\frac{1}{2(n-1)}R_gg),\\\\\n&W_g\\doteq \\mathrm{Riem}_g - \\frac{1}{n-2}\\overset{\\circ}{\\mathrm{Ric}}\\owedge g - \\frac{R_g}{2n(n-1)}g\\owedge g,\\\\\n&h\\owedge k(X,Y,Z,V)\\doteq h(X,Z)k(Y,W) + h(Y,V)k(X,Z) - h(X,V)k(Y,Z) - h(Y,Z)k(X,V),\\\\\n&T_g\\doteq (n-2)(\\nabla^2\\mathrm{tr}_gS_g - \\frac{g}{n}\\Delta_g\\mathrm{tr}_gS_g) + 4(n-1)(S\\times S - \\frac{1}{n}|S_g|^2_g g) - n\\mathrm{tr}_g\\mathrm{S}_g \\overset{\\circ}{S}_g,\\\\\n&(S\\times S)_{jk}\\doteq S^i_jS_{ik},\n\\end{split}\n\\end{align*}\nwhere in the above we have defined several useful tensor and operations, namely, the Bach tensor $B_g$; the Cotton tensor $C_g$; the Schouten tensor $S_g$; the Weyl curvature tensor $W_g$, which uses the $\\owedge$-operator, which for two $(0,2)$-tensor fields $h$ and $k$ produces a $(0,4)$-tensor field. Since the Bach tensor is known to be traceless, and from the above expressions we have that $\\mbox{tr}_gT_g=0$, then $\\mathrm{tr}_g J_g=Q_g$. Taking into consideration all this notation, we are now in a position to present the following result, whose proof runs in complete analogy to the previous ones.\n\n\\begin{coro}\nLet $(M^n,g_0)$ is an $H_{s+3,\\delta_0}$-asymptotically euclidean manifold without boundary, with $n\\geq 3$, $\\delta>-\\frac{n}{2}$, $\\delta_0\\geq-2$ and $s>\\frac{n}{2}$. Then, there is a constant $C$, such that for any $H_{s+5,\\delta}$-AE metric $g$, with $\\delta>-\\min\\{\\frac{n}{2}+2,4\\}$, the following inequality holds\n\\begin{align}\n||Q_g||_{L^2}\\leq C ||\\overset{\\circ}{J_g}||_{L^2}\n\\end{align}\n\\end{coro}\n\\begin{proof}\nWe basically only need to show that $J_g$ and $Q_g$ satisfy the hypotheses of Lemma \\ref{B-AS}. Since $g$ is $H_{s+5,\\delta}$-AE with $\\delta>-\\frac{n}{2}$, then from the continuous multiplication property and the fact that $\\mathrm{Riem}_g, \\mathrm{Ric}_g\\in H_{s+3,\\delta+2}$, we have that $W_g,S_g\\in H_{s+3,\\delta+2}$. Thus, since contractions define continuous operations on tensor fields, the multiplication property gives us that the contracted tensor ${W_g}_{\\cdot}S\\doteq {W_g}_{ijkl}S^{il}$ is in $H_{s+3,\\delta+4}$. Also, we get that $C_g\\in H_{s+2,\\delta+3}$, which, together with ${W_g}_{\\cdot}S\\in H_{s+3,\\delta+4}$, gives us $B_g\\in H_{s+1,\\delta+4}$. Similarly, we have that $\\Delta_gR_g\\in H_{s+1,\\delta+4}$ and $R^2_g,|\\mathrm{Ric}_g|^2_g\\in H_{s+3,\\delta+4}$, which gives us that $Q_g\\in H_{s+1,\\delta+4}$. Finally, the fact that $S\\in H_{s+3,\\delta+2}$ and the multiplication property give us that $T\\in H_{s+1,\\delta+4}$. All this implies that $Q_g,J_g\\in H_{s+1,\\delta+4}$, where $\\rho=\\delta+4\\geq 0$ and $\\rho>-\\frac{n}{2}+2$, since $\\delta>-\\min\\{\\frac{n}{2}+2,4\\}$. Thus, we are under the hypotheses of Lemma \\ref{B-AS}, and the result follows.\n\\end{proof}\nFinally, it is worth noticing that Lemma \\ref{B-AS} has been stated in a quite general form, that, as shown above, can be adapted to analyse different versions of almost-Schur type inequalities for different interesting geometric tensors related to conserved quantities. Obviously, we cannot exhaust all the examples here, but, surely, there are other rather obvious candidates for such examples, such us Lovelock curvatures, which have been in the center of plenty of resent research in geometric analysis.\n\n\\subsection*{Static potentials}\n\nRecall that a Riemannian metric $g$ on a manifold $M$ is called static if the linearised scalar curvature map has nontrivial cokernel. This is equivalent to stating that the equation\n\\begin{align}\n-g(\\Delta_gf)+\\nabla^2f-fRic_g=0\n\\end{align}\nadmits some nontrivial solution $f$. Such solution is referred to as a \\textit{static potential}. It has been shown in \\cite{Corvino} that any $C^3$-static metric must have constant scalar curvature, which implies that any $C^3$-asymptotically-flat static metrics must have zero scalar curvature. This means that a static potential of an asymptotically flat static metric must satisfy the system\\footnote{Notice that this means that no static potential can exist in $H_{s,\\delta}$ for $\\delta>-\\frac{n}{2}$, since $\\Delta_g$ is injective in this case.}\n\\begin{align}\\label{staticeq}\n\\begin{split}\n\\nabla^2f&=fRic_g,\\\\\n\\Delta_gf&=0.\n\\end{split}\n\\end{align}\n\nPositive static potentials are an interesting object of study, since they are intimately related with vacuum space-time solutions of the Einstein equations, as was shown in \\cite{Corvino}. In fact, there it is shown that a Riemannian manifold $(M^n,g)$ admits a static potential iff the warped product metric $\\tilde{g}=-f^2dt\\otimes dt + g$ is Einstein. Thus, in the case of $(M,g)$ being an AE manifold, since $R(g)=0$, then $\\tilde{g}$ must actually be Ricci-flat. Thus, we get a correspondence between \\textit{static} vacuum solutions of the space-time Einstein equations with static Riemannian manifolds. Furthermore, in \\cite{Corvino} it is also shown that the zero level set of a static potential (in case it exists), is given by a regular totally geodesic hypersurface in $M$. Notice that in case $f^{-1}(0)\\neq \\emptyset$, then, the space-time metric $\\tilde{g}$ will actually degenerate on such set, possibly signalling the existence of some \\textit{pathology} of the space-time structure. In fact, since the scalar curvature of a static Riemannian metric is zero, then it is trivially a solution of the \\textit{time-symmetric} vacuum Einstein constraint equations, which under evolution generate the static space-time. In this scenario, the hypersurface $f^{-1}(0)$ defines an \\textit{apparent horizon} in the initial data. Such structures are related with the formation of black holes under the space-time evolution. For some details concerning these ideas, see, for instance, \\cite{maxwell}-\\cite{Corvino}-\\cite{chrusciel-mazzeo} and references therein. \n\nTaking into account the above ideas, it is not surprissing that static potentials in the context of AE manifolds have atractted quite a lot of attention, and some strong rigidity has been observed for instance in \\cite{Miao} for the $3$-dimensional case. In this section, we intend to show how some higher-dimensional analogues of their results follow naturally from the Pohozaev-Schoen identity (\\ref{PS}), plus some mild hypotheses on $f$. Before going to the main statements, we make explicit the fact that throughout this section, when we consider static AE manifolds with boundary, we will consider that $\\partial M=f^{-1}(0)$, this being motivated by the above discussion. That this, we have a model in mind where the boundary components would signal the existence of apparent horizons in time-symmetric vacuum initial data sets for the Einstein equations, which, under evolution, should evolve into black hole static space-times. The following lemma, which can be derived using some results that can be found \\cite{Miao}, is presented within our current notations and conventions. We include the proof for sake of completeness.\n\n\\begin{lemma}\nSuppose that $f$ is a static potential of an $H_{s+3,\\delta}$-asymptotically euclidean metric $g$, with $s>\\frac{n}{2}$, $\\delta>-1$, and $n\\geq 3$. Then $f\\in C^2$ has bounded gradient $\\nabla f$ and $\\nabla^2f\\in L^2$.\n\\end{lemma}\n\\begin{proof}\nFirst of all, concerning the regularity of the solution, since $H_{s+3,\\delta}$ embeds in $C^3$, appealing to Proposition 2.5 in \\cite{Corvino}, we get that the static potential is $C^2$. Now, consider a point $x\\in M$ which lies in one of the ends $E_i$. Let $B_{r_0}(p)$ be the $g$-geodesic ball of radius $r_0$ around the origin $p$, with $r_0$ chosen large enough such that its boundary $\\partial B_{r_0}$ lies in the ends of $M$, and, furthermore, suppose that $x$ has been chosen so that it lies outside this geodesic ball. Let $\\gamma(t)$ be a $g$-minimizing geodesic joining $p$ and $x$, parametrized by arc-length, so that $\\gamma(r_0)\\in\\partial B_{r_0}$, and suppose that $\\gamma(T)=x$. Finally, consider $f(t)\\doteq f(\\gamma(t))$, $r_0\\leq t\\leq T$. Then, since $f$ is a static potential of $g$, we have\n\\begin{align}\\label{miao1}\nf''(t)=\\mathrm{Ric}_g(\\gamma',\\gamma')f(t) \\;\\;\\ \\forall \\;\\; r_0\\leq t\\leq T\n\\end{align}\nSince $g-e\\in H_{s+2,\\delta}$, then $\\mathrm{Ric}_g\\in H_{s,\\delta+2}\\subset L^2_{\\delta+2}\\cap C^0$, which implies that $|\\mathrm{Ric}_g|_e=o(d_e(x)^{-(\\delta+2+\\frac{n}{2})})$ near infinity, in each end. The asymptotic condition on $g$ actually gives us that $|\\mathrm{Ric}_g|_e=o(d_g(x)^{-(\\delta+2+\\frac{n}{2})})$. This implies that, given an arbitrary $\\epsilon>0$, if $t$ is sufficiently large, then \n\\begin{align}\\label{miao2}\n|\\mathrm{Ric}_g(\\gamma'_t,\\gamma'_t)|(\\gamma_t)\\leq \\epsilon d_g(\\gamma_t)^{-(\\delta+\\frac{n}{2}+2)}\\leq \\epsilon t^{-2}.\n\\end{align}\n\nNow, define $\\alpha\\doteq \\frac{1}{2}(1+\\sqrt{1+4\\epsilon})$; $a\\doteq \\sup_{\\partial B_{r_0}}(|f|+|Df|_e)$ and $\\omega(t)\\doteq At^{\\alpha}$, with $r_0\\leq t\\leq T$, where $A$ is a constant chosen such that $Ar_0^{\\alpha}>a$ and $\\alpha Ar_0^{\\alpha-1}>a$ and $\\epsilon$ is some positive constant. Then, the function $\\omega(t)$ satisfies the following properties\n\\begin{align}\\label{miao3}\n\\omega''(t)=\\epsilon t^{-2}\\omega(t), \\;\\; |f(r_0)|\\leq \\omega(r_0), \\;\\; |f'(r_0)|\\leq \\omega'(r_0).\n\\end{align}\n\nSuppose that $|f(t)|>\\omega(t)$ for some $r_0\\leq t\\leq T$. Then, define $t_1\\doteq \\inf \\{t\\in [r_0,T] \\;\\; \\backslash \\;\\; |f(t)|>\\omega(t) \\}$. We know that $t_1$ satisfies that $t_1>r_0$ and $f(t_1)=\\omega(t_1)$. Using (\\ref{miao1})-(\\ref{miao2}), we have that \n\\begin{align*}\n|f''(t)|\\leq \\epsilon t^{-2}\\omega(t)=\\omega''(t) \\;\\; \\forall \\;\\; r_0\\leq t\\leq t_1.\n\\end{align*}\nNow, integrating he above inequality twice and using (\\ref{miao3}), gives $|f(t)|\\leq \\omega$ for all $r_0\\leq t\\leq t_1$, which is a contradiction, thus we get that $|f(t)|\\leq \\omega(t)$ for all $t\\in [r_0,T]$. Thus, since $\\delta>-\\frac{n}{2}$, we can choose an $\\epsilon>0$ such that $\\alpha<1+\\frac{1}{2}(\\delta+\\frac{n}{2})$, which implies that\n\\begin{align*}\n|f(t)|\\leq At^{1+\\frac{1}{2}(\\delta+\\frac{n}{2})}.\n\\end{align*}\nThe above inequality together with (\\ref{miao1})-(\\ref{miao2}), gives that\n\\begin{align*}\n|f''(t)|\\leq A\\epsilon t^{-(\\delta+2+\\frac{n}{2})}t^{1+\\frac{1}{2}(\\delta+\\frac{n}{2})}= A\\epsilon t^{-(1+\\frac{1}{2}(\\delta+\\frac{n}{2}))} \\;\\; \\forall \\;\\; r_0\\leq t \\leq T.\n\\end{align*}\nThis implies that there is a constant $C_1$, independent of the point $x\\in E_i\\backslash B_{r_0}$ and of $t$, such that \n\\begin{align}\\label{miao4}\n|f''(t)|\\leq C_1 t^{-(1+\\frac{1}{2}(\\delta+\\frac{n}{2}))} \\;\\; \\forall \\;\\; r_0\\leq t \\leq T.\n\\end{align}\nIntegrating this inequality between $r_0$ and $t$, we can show that $|f'(t)|\\leq C_2$ for some other constant $C_2$, which shows that for sufficiently large $|x|$ it holds that\n\\begin{align}\n|f|(x)\\leq C_3|x|.\n\\end{align}\n\nNow, using the equation $\\nabla^2f=f\\mathrm{Ric}_g$, we have that $|\\nabla^2f|_e(x)\\lesssim |x||x|^{-(\\delta+2+\\frac{n}{2})}=|x|^{-(1+\\delta+\\frac{n}{2})}\\in L^2(\\mathbb{R}^n\\backslash B_1(0))$ for $\\delta>-1$, which proves that $\\nabla^2f\\in L^2$. In order to prove that $\\nabla f$ is bounded, define $\\phi\\doteq|\\nabla f|^2_g$, and notice that \n\\begin{align*}\n|\\nabla\\phi|^2_g(x)\\leq 4|\\nabla^2f|^2_g\\phi \n\\end{align*}\nDoing as above, consider $\\phi(t)\\doteq \\phi(\\gamma(t))$, and notice that since $g$ is AE, then $g$ and $e$ define \\textit{equivalent} metrics, that is, for any tangent vector $v_p$ to $M$, there are positive constants $a$ and $b$, independent of the point $p\\in M$, such that $a|v_p|_e\\leq |v_p|_g \\leq b|v|_e$. Thus, the above inequality tells us that $|\\nabla\\phi|^2_e(x)\\lesssim |\\nabla^2f|^2_e\\phi$. Then, we have that\n\\begin{align*}\n|\\phi'(t)|=|g(\\nabla\\phi_{\\gamma(t)},\\gamma'_t)|\\leq |\\nabla\\phi|_g\\leq 2|\\nabla^2f|_g\\phi^{\\frac{1}{2}} \\lesssim |\\nabla^2f|_e\\phi^{\\frac{1}{2}} \\lesssim t^{-(1+\\delta+\\frac{n}{2})}\\phi^{\\frac{1}{2}}(t),\n\\end{align*}\nwhich, after integration, gives us that\n\\begin{align*}\n\\frac{1}{\\delta+\\frac{n}{2}}\\left( t^{-(\\delta+\\frac{n}{2})} - r^{-(\\delta+\\frac{n}{2})}_0\\right)\\lesssim \\phi^{\\frac{1}{2}}(t) -\\phi^{\\frac{1}{2}}(r_0) \\lesssim -\\frac{1}{\\delta+\\frac{n}{2}}\\left( t^{-(\\delta+\\frac{n}{2})} - r^{-(\\delta+\\frac{n}{2})}_0\\right).\n\\end{align*}\nSince $t> r_0$ and $\\delta+\\frac{n}{2}>0$, the above inequality implies that\n\\begin{align*}\n\\phi^{\\frac{1}{2}}(r_0)-\\frac{1}{\\delta+\\frac{n}{2}} r^{-(\\delta+\\frac{n}{2})}_0\\lesssim \\phi^{\\frac{1}{2}}(t) \\lesssim \\phi^{\\frac{1}{2}}(r_0)+ \\frac{1}{\\delta+\\frac{n}{2}} r^{-(\\delta+\\frac{n}{2})}_0 \\;\\;\\; \\forall \\;\\;\\; t> r_0.\n\\end{align*}\nThe above relation implies that $|\\nabla f|_g(x)=\\phi^{\\frac{1}{2}}(x)$ is bounded.\n\n\\end{proof}\n\n\n\\begin{thm}\nSuppose that $(M^n,g)$ is a $H_{s+3,\\delta}$-AE manifold with $n\\geq 3$, $s>\\frac{n}{2}$ and $\\delta>-1$, which admits a non-negative static potential $f$. Furthermore, suppose that $\\mathrm{Ric}_g\\in H_{s+1,\\rho}$, for some $\\rho>\\frac{n}{2}-1$, and that $\\partial M=f^{-1}(0)$ consists of $N$ closed connected components, labelled by $\\{\\Sigma_i \\}_{i=1}^{N}$. Then, it follows that \n\\begin{align}\n\\int_Mf|\\mathrm{Ric}_g|^2_g\\mu_g = \\frac{1}{2}\\sum_{i=i}^Nc_i\\int_{\\Sigma_i}R_{h_i}\\mu_{\\partial M},\n\\end{align}\nwhere the constants $c_i=|\\nabla f|_{\\Sigma_i}$. In particular, if $\\partial M=\\emptyset$, then $(M^n,g)$ is isometric to $(\\mathbb{R}^n,e)$, where $e$ is the euclidean metric.\n\\end{thm}\n\\begin{proof}\nFirst, notice that under our hypotheses the above lemma shows that $X=\\nabla f$ satisfies the general hypotheses of Theorem \\ref{thm1} if we pick $\\rho>\\frac{n}{2}-1$. Also, since a static AE metric has zero scalar curvature, we get that $\\mathrm{Ric}_g\\in H_{s+1,\\rho}$ is a conserved $(0,2)$-tensor field. Thus, our choice of $\\rho>\\frac{n}{2}-1$ shows that we are under the hypotheses of Theorem \\ref{thm1}. Then, we can apply (\\ref{PS}) choosing the vector field $X=\\nabla f$ so as to get\n\\begin{align}\\label{SP1}\n\\int_M\\langle \\mathrm{Ric}_g,\\nabla^2f \\rangle\\mu_g=\\int_{\\partial M}\\mathrm{Ric}(\\nabla f,\\nu)\\mu_{\\partial M}.\n\\end{align}\nAlso, from the static equation, we have that $\\langle \\mathrm{Ric}_g,\\nabla^2f \\rangle=f|\\mathrm{Ric}_g|^2_g$. Furthermore, since $\\partial M=f^{-1}(0)$, then, we get that $\\nu=-\\frac{\\nabla f}{|\\nabla f|}$. In addition, the static equation implies that on $\\partial M$ it holds that $\\nabla^2f(X,\\nabla f)=0$ for any $X$ tangent to $\\partial M$. Then, since $\\nabla^2f(X,\\nabla f)=\\frac{1}{2}X(|\\nabla f|^2_g)$, we get that $|\\nabla f|$ is constant along each connected component of $\\partial M$. Thus, we see that\n\\begin{align*}\n\\int_Mf|\\mathrm{Ric}_g|^2_g\\mu_g = -\\sum_{i=i}^Nc_i\\int_{\\Sigma_i}\\mathrm{Ric}(\\nu,\\nu)\\mu_{\\partial M},\n\\end{align*}\nwhere the sum is carried along the $N$ connected components $\\{\\Sigma_i\\}_{i=1}^N$ of $\\partial M$, and the constants $c_i=|\\nabla f|_{\\Sigma_i}$. Now, denote by $h_i$ the induced Riemannian metric on $\\Sigma_i$. Now, let $\\{E_i,\\nu\\}_{i=1}^{n-1}$ denote an orthonormal frame of $M$ along $\\Sigma_i$, where $\\{ E_i \\}_{i=1}^{n-1}$ gives an orthonormal frame on $\\Sigma_i$. Then, we get that for any pair of tangent vectors $X,Y$ to $\\Sigma_i$ it holds that\n\\begin{align*}\n\\mathrm{Ric}_g(X,Y)&=\\sum_{i=1}^{n-1}g(R_g(X,E_i)E_i,Y) + g(R_g(X,\\nu)\\nu,Y),\n\\end{align*}\nwhich, from the Gauss equation and the fact that all the components are totally geodesic, implies that\n\\begin{align*}\nR_g&=\\sum_{j=1}^{n-1}\\mathrm{Ric}_g(E_j,E_j) + \\mathrm{Ric}_g(\\nu,\\nu)=\\sum_{j=1}^{n-1}\\sum_{i=1}^{n-1}g(R_g(E_j,E_i)E_i,E_j) + \\sum_{j=1}^{n-1}g(R_g(E_j,\\nu)\\nu,E_j) \\\\\n&+ \\mathrm{Ric}_g(\\nu,\\nu),\\\\\n&=\\sum_{j=1}^{n-1}\\sum_{i=1}^{n-1}h(R_{h_i}(E_j,E_i)E_i,E_j) + \\sum_{j=1}^{n-1}g(R_g(\\nu,E_j)E_j,\\nu) + \\mathrm{Ric}_g(\\nu,\\nu),\\\\\n&=R_{h_i} + 2\\mathrm{Ric}_g(\\nu,\\nu).\n\\end{align*}\nAgain, since $R_g=0$, we get $-\\mathrm{Ric}_g(\\nu,\\nu)|_{\\Sigma_i}=\\frac{1}{2}R_{h_i}$. Thus, going back, this gives us that\n\\begin{align}\\label{static1}\n\\int_Mf|\\mathrm{Ric}_g|^2_g\\mu_g = \\frac{1}{2}\\sum_{i=i}^Nc_i\\int_{\\Sigma_i}R_{h_i}\\mu_{\\partial M}.\n\\end{align}\n\nFrom the above equation it is clear that if $\\partial M=\\emptyset$, then $\\mathrm{Ric}_g=0$ on $M$. In such case, we have already shown this implies that $(M,g)$ has to be isometric to $(\\mathbb{R}^n,e)$.\n \n\\end{proof}\n\n\\begin{remark}\nNotice that the two conditions $g-e\\in H_{s+3,\\delta}$, with $\\delta>-1$, and $\\mathrm{Ric}_g\\in H_{s+1,\\rho}$, with $\\rho>\\frac{n}{2}-1$, in dimensions 3 and 4 are actually redundant, since, from $g-e\\in H_{s+3,\\delta}$, we get that $\\mathrm{Ric}_g\\in H_{s+1,\\delta+2}$, with $\\delta+2>1$. Thus, in dimension $n=3$, we get $\\frac{n}{2}-1=\\frac{1}{2}<\\delta+2$. Similarly, for $n=4$, we get $\\frac{n}{2}-1=1<\\delta+2$. Nevertheless, for $n\\geq 5$ the condition on the Ricci tensor becomes a necessary additional information. \n\nFrom the above discussion, we see that the identity (\\ref{static1}) is presented for $n=3$ in \\cite{Miao}, and that the same identity holds with the same hypotheses for $n=4$. Nevertheless, for $n\\geq 5$ the identity extends naturally, although non-trivially. \n\\end{remark}\n\n\n\\bigskip\nNow, consider a connected static manifold $(M^n,g,f)$, with $n\\geq 3$, where $g$ and $\\mathrm{Ric}_g$ satisfy the hypotheses of the above Lemma, and suppose that $\\partial M=\\emptyset$. Notice that if $f$ changes sign, then $f^{-1}(0)\\neq \\emptyset$ actually represents the boundary of both $f^{-1}(0,\\infty)$ and $f^{-1}(-\\infty,0)$, which are two $n$-dimensional submanifolds of $M$. We can then add $f^{-1}(0)$ to any of these submanifolds, generating two manifolds with boundary, say $M_{+}$ and $M_{-}$ respectively. Clearly, $(M_{+},g,f)$ is a static manifold with boundary which satisfies all the hypotheses of the above theorem provided suitable decaying conditions for $g$. Furthermore, $(M_{-},g,-f)$ is also a static manifold since the static equation is linear. Thus, $(M_{-},g,-f)$ also satisfies all the hypotheses of the above theorem. Thus, we get the following corollary.\n\n\\begin{coro}\nLet $(M^n,g,f)$ be a connected static manifold, with $n\\geq 3$, where $g$ and $\\mathrm{Ric}_g$ satisfy the hypotheses of the above theorem. Moreover, suppose that $f^{-1}(0)\\neq \\emptyset$ and $\\partial M=\\emptyset$. Then, we have that\n\\begin{align}\\label{balckholetopology.1}\n\\sum_{i=i}^Nc_i\\int_{\\Gamma_i}R_{h_i}\\mu_{\\partial M} \\geq 0,\n\\end{align}\nwhere $\\{\\Gamma_i\\}_{i=1}^N$ denote the connected components of $f^{-1}(0)$. Furthermore, the equality holds if and only if $(M,g)$ is isometric to $(\\mathbb{R}^n,e)$.\n\\end{coro}\n\\begin{proof}\nConsider the notation introduced above. Then, under our hypotheses, we have that both $(M_{+},g,f)$ and $(M_{-},g,-f)$ are static manifolds with boundary, for which the above theorem holds. Then, we get that\n\\begin{align*}\n\\sum_{i=i}^Nc_i\\int_{\\Gamma_i}R_{h_i}\\mu_{\\partial M}=2\\int_{M_{+}}f|\\mathrm{Ric}_g|^2_g\\mu_g\\geq 0,\\\\\n\\sum_{i=i}^Nc_i\\int_{\\Gamma_i}R_{h_i}\\mu_{\\partial M}=-2\\int_{M_{-}}f|\\mathrm{Ric}_g|^2_g\\mu_g \\geq 0,\n\\end{align*}\nwhere, in both cases, the equality holds iff $\\mathrm{Ric}_g=0$ on both $M_{+}$ and $M_{-}$. That is, the equality holds iff $\\mathrm{Ric}_g=0$ on $M$. But then, from Theorem \\ref{Ricciflatrigidity}, we get the rigidity statement in the equality case.\n\\end{proof}\n\nFrom the above corollary, we get the following.\n\n\\begin{coro}\nUnder the same hypotheses of the above corollary, if $n=3$ and $f^{-1}(0)$ consists of one connected component, that is $N=1$, then $f^{-1}(0)$ is homeomorphic to $\\mathbb{S}^2$. If $N\\geq 2$, then there must be at least one topological sphere within the components of $f^{-1}(0)$.\n\\end{coro}\n\n\\begin{proof}\nThis is straightforward, since, from (\\ref{balckholetopology.1}), we get that if $N=1$ the total scalar curvature of $\\partial M$ must be positive. Thus, from the Gauss-Bonnet theorem, we know that the Euler characteristic of $\\partial M$ must be positive, thus $\\partial M$ is a topological $2$-sphere.\n\nOn the other hand, in the case $N\\geq 2$, since the sum of the total scalar curvatures of the different components has to be positive, then we must have at least one sphere from the argument given above.\n\\end{proof}\n\n\\begin{remark}\nIt is important to stress that the above corollary is very closely related to well-known results related with the classification of allowable black hole topologies. Recall that the zero level set of a static potential models the intersection of the event horizon of a static black hole with a $t$-constant hypersurface in space-time. Thus, the characterization given above shows that such slices of the event horizon of an isolated static black hole have to be a topological 2-spheres. Once this connection between the two problems is clear, it also becomes clear that the above corollary is not a new result, but something that can be extracted from more general statements, such as Hawking's black hole uniqueness theorem \\cite{Hawking}, or even more generally, a recent result by Eichmair, Galloway and Pollack \\cite{EGP}. In fact, these results are much stronger than the above corollary, since the show that every component of $\\partial M$ should be a topological $2$-sphere and they are not restricted to static space-times. Nevertheless, we consider that the value of the above corollary relies on its simplicity, since this result is a straightforward application of the generalized Pohozaev-Schoen identity, which, in turn, relies only on a conservation identity derived from a symmetry principle. In contrast, the sharper characterizations presented in \\cite{Hawking} and \\cite{EGP} rely on more delicate techniques. In particular, for instance, the result presented in \\cite{EGP} relies on the positive resolution of the geometrization conjecture. \t\n\\end{remark}\n\n\\section*{Acknowledgements}\n\nThe first author would like to thank professor Paul Laurain for reading a previous version of this paper and making several relevant comments and suggestions.\n \nThe first author would also like to thank CAPES\/PNPD for financial support.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nThe axion\\cite{Peccei,Wilczek,Kim,DFS} is the Nambu-Goldstone boson\nassociated with the Peccei-Quinn symmetry breaking which was invented\nas the most natural solution to the strong CP problem\\cite{tHoot} of QCD.\nThe Peccei-Quinn symmetry breaking scale $F_a$ is stringently\nconstrained by the consideration of accelerator experiments, stellar\ncooling and cosmology. The allowed range of $F_a\/N$ ( axion window ) is\nbetween $10^{10}\\mbox{GeV}$ and $10^{12}\\mbox{GeV}$, where the integer $N$\nis the color anomaly of Peccei-Quinn symmetry. \n\nSince the Peccei-Quinn symmetry is a global U(1) symmetry, global\nstrings (axionic strings) are produced during spontaneous symmetry\nbreaking. At this\nstage the potential for the axion field is flat, i.e., the axion is\nmassless. However the axion has a mass at QCD scale\nthrough the instanton effect. \nThe potential is written by\n\\begin{equation}\n V(A) = f_{\\pi}^2m_{\\pi}^2\\left( 1 - \\cos\\frac{NA}{F_a}\\right),\n \\label{potential}\n\\end{equation}\nwhere $f_{\\pi}$ is the pion decay constant and $m_{\\pi}$ is its mass.\nThen the mass of the axion is given by $m_a \\simeq\nf_{\\pi}m_{\\pi}\/{F_a\/N}$. This potential(\\ref{potential}) has $Z_n$\nsymmetry and takes its minimum at $A = 0, (f_a\/N)\\pi, 2(f_a\/N)\\pi,\n\\ldots 2f_a\\pi$. $Z_n$ discrete symmetry is spontaneously broken to\nproduce two dimensional topological defects, i.e., axionic domain walls.\nThe property of the axionic domain wall is characterized by $N$.\nSince $N$ domain walls stretch out from each axionic string, the network\nof the string-wall systems is very complicated and survives long enough\nto dominate the universe for $N > 1$ axionic domain wall \\cite{RPS}.\nTherefore the $N>1$ domain wall is not accepted cosmologically.\n\nOne might expect that the density of the domain wall can be diluted in\nthe inflationary universe. However, for the dilution mechanism to\nwork, both the reheating temperature and the expansion rate during\ninflation should be smaller than $F_a$ \\cite{Linde}, which is rather\nunnatural requirement for many inflation models. Therefore the domain\nwall problem in the axion model is serious one.\n\nIn the case of $N=1$, the domain wall is a disk bounded by the axionic\nstring. The wall with the string boundary is no longer stable and it\nmight collapse by the surface tension. In fact, since, as seen later,\nthe surface tension is stronger than the tension due to the string for\nthe domain wall with size much greater than the width of the wall, the\ndynamics of the wall is controlled by the wall tension. Therefore the\nstring that bounds the wall cannot prevent the wall from collapsing.\nTypically the wall has size of horizon length ($\\sim t_{QCD}$) when it\nis formed. Therefore the time scale for collapse is about $t_{QCD}\n\\sim 10^{-4}$ sec and the walls disappear quickly without overclosing\nthe universe and hence the domain wall seems harmless in the case of\n$N=1$ axion model. However it is expected that a number of axions are\nproduced when the axionic walls collapse. If produced axions are\nnon-relativistic or cold, their energy density decreases slowly\n($\\propto a^{-3}$ $a$: scale factor) compared with the radiation\ndensity ($ \\propto a^{-4}$). Hence the relative contribution of axions\nto the total density of the universe increases with $a$ until the\nuniverse becomes matter dominated.\n\nIn this paper we study the collapse of the axionic domain wall ($N=1$)\nby the numerical integration of equations of motion for the axion\nfield and estimate the energy density of axion field after the\ncollapse. It is found that the energy carried by the walls is\nconverted to the axions, which gives significant contribution to the\nmass density of the present universe and might account for the dark\nmatter of the universe.\n\nThe axion emission is also expected from axionic string before\n$t_{QCD}$\\cite{Davis} or annihilation of axionic domain\nwalls\\cite{Lyth}. However, since it is shown later that parallel two\ndomain walls go through each other without annihilation, it is\nunlikely that a large number of axions are produced by interaction of\ntwo domain walls. The massless axion can be produced by axionic\nstrings and they acquire mass after the QCD phase transition, which\nmight give significant contribution to the total density of the\npresent universe. However there are two independent quantitative\nestimations by Davis\\cite{Davis} and by Harari and\nSikivie\\cite{Hagmann} and, unfortunately, they are quite different.\nThe contribution from axionic domain walls is expected to be at least\ncomparable to that from axionic strings if the estimation by Harari\nand Sikivie is correct.\n\n\\section{Dynamics of Domain Wall}\n\nThe dynamics of the Peccei-Quinn scalar field $\\phi$ is described by\nthe lagrangian:\n\\begin{eqnarray}\n {\\cal L} & = & \\partial^{\\mu} \\phi^{*} \\partial_{\\mu} \\phi \n + \\frac{\\lambda}{4} [|\\phi|^2 - F_a^2]^2\n \\nonumber\\\\\n & & = \\partial^{\\mu} \\phi^{*} \\partial_{\\mu} \\phi + V_s(|\\phi|)\n \\label{eq:lagra}\n\\end{eqnarray}\nwhere $\\lambda$ is the coupling constant. As the universe cools down,\nPeccei-Quinn U(1) symmetry ($\\phi \\rightarrow e^{i\\theta}\\phi$) is\nspontaneously broken and the scalar field has vacuum expectation value\n$\\langle | \\phi | \\rangle = F_a$. After global U(1) symmetry is\nbroken, axionic strings are formed. From causality argument, about one\naxionic string is produced within the horizon at $T\\simeq F_a$. Since\nthe line energy density of the string is $4\\pi F_a^2$, the string\ndensity $\\rho_{st}$ is about $4\\pi F_a^2 t\/t^3 \\sim 4\\pi\nF_a^4(F_a\/m_{pl})^2$ which is only $\\sim 10^{-13} - 10^{-17}$ of total\ndensity of the universe at the formation epoch. Defining\n$\\phi = |\\phi|\\exp(iA\/F_a)$, the lagrangian for axion field\n$A$ is derived from eq.(\\ref{eq:lagra}) as\n\\begin{eqnarray}\n {\\cal L}_a & = & \\partial^{\\mu} A \\partial_{\\mu} A \n + f_{\\pi}^2m_{\\pi}^2\\left( 1 - \\cos\\frac{A}{F_a}\\right) ,\n \\nonumber\\\\\n & & = \\partial^{\\mu} A \\partial_{\\mu} A + V_w(A)\n \\label{eq:alagra}\n\\end{eqnarray}\nwhere the second term comes from QCD instanton effect which gives\naxion mass $m_a \\simeq f_{\\pi}m_{\\pi}\/{F_a}$. The potential $V_w(A)$\nhas a minimum at $A = 0, 2\\pi F_a$ and domain walls are produced\nbetween $A=0$ phase and $A = 2\\pi F_a$ phase. More precisely, the\naxion mass has temperature dependence and increase as \n\\begin{equation}\n m_a(T) \\simeq 0.1m_a(T=0) (\\Lambda_{QCD}\/T)^{3.7},\n\\end{equation}\nwhere $\\Lambda_{QCD}$ is the QCD scale $\\sim 200\\mbox{MeV}$. The domain\nwalls are produced when the axion mass becomes greater than the\nexpansion rate of the universe, i.e., $m_a(T_1) \\simeq\n\\dot{a}(T_1)\/a$. The axionic domain wall has size of about horizon\nlength at the formation time $t_1(T_1)$ and has the axionic string on\nits boundary. The surface tension of the axionic domain wall is\n$\\sigma \\simeq 16 m_a F_a^2$\\cite{Vilenkin}. For wall with size $R$,\nthe ratio of surface energy to string energy is given by\n\\begin{equation}\n \\frac{\\mu R}{\\sigma R^2} = \\frac{4\\pi F_a^2}{16m_a F_a^2 R} \\simeq\n \\frac{1}{m_a R}.\n \\label{walltostring}\n\\end{equation}\nTherefore the dynamics of the axionic wall with size greater than\n$R^{*} \\simeq 1\/(m_a) \\gg t_1$ is determined by the wall tension and\nthe string tension can be neglected.\n\nThe cosmological evolution of the wall is determined by the surface\ntension and interaction of walls; the former makes the wall shrink and\nthe latter cuts the wall into small pieces. In both cases the wall\nfinally collapses by the surface tension. After collapse the domain\nwalls disappear and the energy that the wall had is converted into\nkinetic energy of axion fields. If the energy of axion fields changes\nlike non-relativistic particles as the universe expands, the axion\nenergy density is \n\\begin{equation}\n \\rho_a(t) = \\rho_{wall}(t_1)\\left(\\frac{a(t_1)}{a(t)}\\right)^3,\n \\label{axiondensity}\n\\end{equation}\nwhere $\\rho_a(t)$ and $\\rho_{wall}$ are the energy density of the\naxion field and the axionic wall, respectively. For relativistic\naxion, the axion density decreases as $\\sim a^{-4}$ until axion\nbecomes non-relativistic. Then eq.(\\ref{axiondensity}) is changed to\n\\begin{equation}\n \\rho_a(t) = \\rho_{wall}(t_1)\\left(\\frac{a(t_1)}{a(t)}\\right)^3 \n \\left(\\frac{\\langle E_a \\rangle}{m_a}\\right)^{-1},\n \\label{axiondensity2}\n\\end{equation}\nwhere $\\langle E_a \\rangle$ is the average energy of emitted axion at\n$t_1$. Assuming the mean distance and the mean radius of\nwalls are $\\alpha t_1$ and $\\beta t_1$, $\\rho_{wall}$ is given by\n\\begin{equation}\n \\rho_{wall}(t_1) = \n \\frac{\\sigma \\pi (\\beta t_1)^2}{(\\alpha t_1)^3}\n = 16\\pi\\alpha^{-3}\\beta^2 m_a F_a^2\n \\left(\\frac{16\\pi^3 {\\cal N}}{45}\\right)^{1\/2}\\frac{T_1}{m_{pl}}, \n \\label{walldensity}\n\\end{equation}\nwhere ${\\cal N}$ is the relativistic degrees of freedom at $t_1\\simeq 1\n\\mbox{GeV}$. We expect that the numerical parameters $\\alpha$ and $\\beta$\nare O(1) from causality, although the precise values should be determined\nby the realistic simulation of cosmological formation of the wall-string\nsystem, which is beyond the scope of the present paper.\n \nUsing the entropy density of the universe $s (= 2\\pi^2{\\cal N}T^3\/45)$\nand its conservation ( $s a^3 =$ const.),\nthe number density of the axion can be written as\n\\begin{equation}\n \\frac{n_a}{s} = 380 \\alpha^{-3}\\beta^2 {\\cal N}^{-1\/2} \n \\left(\\frac{F_a^2}{T_1 m_{pl}}\\right)\n \\left(\\frac{\\langle E_a \\rangle}{m_a}\\right)^{-1}.\n \\label{axion-entropy}\n\\end{equation}\nThen the present density of the axion is \n\\begin{equation}\n \\rho_a = 1.06 \\times 10^6 \\mbox{cm}^{-3} \\alpha^{-3}\\beta^2\n {\\cal N}^{-1\/2} m_a \n \\left(\\frac{F_a^2}{T_1 m_{pl}}\\right)\n \\left(\\frac{\\langle E_a \\rangle}{m_a}\\right)^{-1}.\n \\label{present-axion-density}\n\\end{equation}\nSince ${\\cal N}=289\/4$, $T_1 \\simeq 2\\mbox{GeV} (F_a\/10\\mbox{GeV})^{-0.18}$\nand $m_a = 6.2\\times 10^{-4}\\mbox{eV}(F_a\/10^{10}\\mbox{GeV})^{-1}$, \n\\begin{equation}\n \\rho_a = 3.2 \\times 10^2 \\mbox{eV} \\mbox{cm}^{-3} \\alpha^{-3}\\beta^2\n \\left(\\frac{F_a}{10^{10}\\mbox{GeV}}\\right)^{1.18}\n \\left(\\frac{\\langle E_a \\rangle}{m_a}\\right)^{-1}.\n\\end{equation}\nThe contribution of the axion to the density parameter $\\Omega$ is given by\n\\begin{equation}\n \\Omega_a h^2 = 0.030 \\alpha^{-3}\\beta^2\n \\left(\\frac{F_a}{10^{10}\\mbox{GeV}}\\right)^{1.18}\n \\left(\\frac{\\langle E_a \\rangle}{m_a}\\right)^{-1},\n \\label{omega-axion}\n\\end{equation}\nwhere $h$ is the Hubble constant in units of $100\\mbox{km}\/\\sec\/\\mbox{Mpc}$.\nTherefore the axion density is large enough to account for the dark\nmatter in the universe unless the axion is ultra-relativistic when it\nis emitted. We estimate the $\\langle E_a \\rangle\/m_a$ by numerical\nsimulation in the next section.\n\n\n\\section{Simulation of Collapse}\n\\label{sec:simulation}\n\nIn order to follow the motion of domain walls, we have solved the\nevolution equation of the axion field numerically. When a wall piece\nmuch smaller than the cosmological horizon is considered, the cosmic\nexpansion can be ignored.\nThen the field equation under the Minkowski background is written as\n\\begin{equation}\n \\frac{\\partial^2 \\phi}{\\partial t^2}-\\nabla^2\\phi\n =-\\frac{~\\partial}{\\partial \\phi}(V_s+V_w)~,\n \\label{eq:eveq}\n\\end{equation}\nusing the potentials in the equations (\\ref{eq:lagra}) and\n(\\ref{eq:alagra}). We have employed the staggered leapfrog method to\nsolve the differential equation. The model parameters are chosen such\nthat the width of the wall is equal to ten simulation meshes and the\nvacuum energy of the string is a hundred times larger than that of the\nwall. The variation of the numerical value in the latter condition\ndoes not alter our conclusion since only the wall tension governs the\nmotion of walls as we have mentioned above. The boundary condition is\nperiodic, which is useful to check the accuracy of our calculation\nsince the total energy in the simulation box is conserved. The\nsolution of a static infinite planar wall under $|\\phi|=v$ is used in\nthe initial configuration.\nWhen the wall lies in $yz-$ plane, it is expressed as\n\\begin{equation}\n \\frac{A(x)}{F_a}=\\pi +2\\sin^{-1}(\\tanh m_a x) ~.\n\\end{equation}\nThus the scale of the inverse axion mass represents the characteristic\nthickness of the axionic wall. The basic numerical technique is\nthe same one in the previous paper.\nSee the reference \\cite{Naga} for more details.\n\nFirst we have confirmed that approaching walls that face in parallel\neach other pass through one another. Widrow showed this is true in the\ncase of a toy sine-Gordon potential\\cite{Widrow}. We have reproduced\nthe passing phenomenon by numerical simulations using the potential\n(\\ref{eq:eveq}). Fig.\\ref{fig:pass} shows the result. The initial\ncondition is set so that the relative velocity is $0.05c$, where $c$\nis light velocity and the separation between walls is 200 meshes.\nNotice that the size of one mesh is equal to $F_a^{-1}$ in our\nsimulation and we take $m_a = 0.1 F_a$. As the time evolves, two\nwalls become close and go away without any crush. When the initial\nrelative velocity is much larger, for example, equal to $0.5c$, the\npair annihilation of walls occurs neither. Therefore it is unlikely\nthat annihilation of domain walls occurs and produces axions\nduring the collision process. The possibility which is described in\nthe reference \\cite{Lyth} should be unfavorable.\n\nOn the other hand, in the case of the encounter of a wall with a wall\nedge, {i.e.}, a string cuts the wall and the process of disintegration\nadvances\\cite{Naga}. Hence by repeated intercommutations, large walls\nare broken to small pieces. When their size becomes comparable with\nthe thickness scale of the wall, they collapse and radiate energy as\naxions. In order to see how fast and effective the wall collapse is,\nwe have performed the simulation of the evolution of a small wall piece.\nAs a simple example we followed the time evolution of a disk wall\nsurrounded by a circular string in the previous work \\cite{Naga}.\nThe result shows that the disk wall shrinks at the\nvelocity of light and the energy of the wall is converted to axions.\nWe can say that the wall collapse process is so rapid that the\nradiation of gravitational waves is hardly expected.\n\nIn order to see the wall collapse with higher accuracy, we have\nperformed the simulations of a strip wall. Figs.\\ref{fig:obi}(a)(b)(c)\ndemonstrate the two-dimensional distribution of $V_w$. The first one\nshows the initial configuration in which there is one strip wall of\ninfinite height and 20 meshes length. As the time evolves, the edges\nof the strip approach each other, that is, the wall size decreases.\nThus the wall collapse proceeds also at about the light velocity and\nthe result of the three-dimensional simulation is confirmed.\nIn the case of Fig.\\ref{fig:obi}, the initial $\\dot{\\phi}$\nis zero everywhere in the box. However, the initial motion may make\nthe wall move periodically so that the death of wall might be avoided.\nWe have found that this is not the case by performing the simulations\nwith various initial motions. Even in the most extreme case where the\nstrings attached to the wall edges go away each other with the light\nvelocity, such a motion could do nothing more than the slight\nextension of the wall life time.\n\nFinally we have estimated the $\\langle E_a \\rangle\/m_a$ in the\nsimulation. The time evolution of the potential energy $V_w$ and\nkinetic energy of the axion field $A$ is shown in\nFig.\\ref{fig:pot}(a). In these simulations, the wall edges are\nsmoothly connected to the true vacuum region which is different from\nthe disk wall case. Since the overestimation of the field gradient is\nremoved by this smoothing, the quantitative analysis of energy\ndistribution is enabled. After the wall collapse there remains only\naxionic waves which we identify as axions and the values of the\npotential and kinetic energy of the axion become constant. Since the\naxion waves oscillate like $\\sim e^{-iE_at}$, the ratio of the kinetic\nenergy to potential energy is equal to $\\langle E_a \\rangle^2\/m_a^2$.\nThe simulation shows $\\langle E_a \\rangle\/m_a \\simeq 3$, which means\nthat the emitted axion is mildly relativistic and becomes\nnon-relativistic soon by cooling due to the cosmic expansion. As\nmentioned before we check the accuracy of our simulation by the\nconservation of the total energy in the simulation box. In the\nsimulation above the total energy is conserved within accuracy of 20\\%.\n\nIn order to confirm the estimated value of $\\langle E_a \\rangle\/m_a $\nwe have performed the simulation of higher resolution ( string\nwidth $= 3$ meshes, wall width $= 30$ meshes, wall length $=60$\nmeshes) . The result (Fig.\\ref{fig:pot}(b)) shows $\\langle E_a\n\\rangle\/m_a \\simeq 3$ which is quite consistent with the result of\nthe lower resolution run.\n\nOur model parameters in the simulations above correspond to $m_a\/F_a =\n0.1$ which is much larger than the actual axion model ( $ m_a\/F_a \\sim\n10^{-23}$ ). To check if the estimation for $\\langle E_a \\rangle\/m_a$\ndepends on $m_a\/F_a$, we have run the simulation of thicker wall\n( string width $= 1$ meshes wall width $= 50$ meshes, wall length $=100$\nmeshes ) in which $m_a\/F_a =0.02$. Although the conservation of the total\nenergy becomes as bad as 50\\% in this case, the result indicates that\nthe value of $\\langle E_a \\rangle\/m_a$ is affected only slightly, which\nmeans the generality of our estimation concerning the relativisity of\nemitted axions.\n\n\\section{Conclusion}\n\\label{sec:conclusion}\n\nIn the previous section, it is found that the axionic wall collapses\nwith time scale $R\/c$ and the mildly-relativistic axions remain\nafter the collapse. The estimated $\\langle E_a \\rangle\/m_a$ is about 3,\nwhich leads to the relic density of the axion given by\n\\begin{equation}\n \\Omega_a h^2 \\simeq 0.01 \\alpha^{-3}\\beta^2\n \\left(\\frac{F_a}{10^{10}\\mbox{GeV}}\\right)^{1.18}\n \\label{omega-axion2}\n\\end{equation}\nSince $\\alpha$ and $\\beta$ are expected to be $O(1)$, the contribution\nof the axion to the present universe is large for $F_a \\gtrsim\n10^{10}\\mbox{GeV}$ (compared with the baryon density $\\Omega_B h^2 \\simeq\n0.013$\\cite{Walker}) and might account for the dark matter of the\nuniverse. However eq.(\\ref{omega-axion2}) should be compared with the\ndensity of the coherent oscillation of axion field (cold\naxion)\\cite{Preskill,Turner} given by\n\\begin{equation}\n \\Omega_a({\\rm cold}) = 6.5\\times 10^{-3\\pm 0.4}\n \\left(\\frac{F_a}{10^{10}\\mbox{GeV}}\\right)^{1.18}.\n \\label{cold}\n\\end{equation}\nTherefore the axion from the wall has comparable to or higher density\nthan the cold axion unless $\\alpha^{-3}\\beta^2$ is much less than 1.\nIn deriving (\\ref{omega-axion2}), we assume that the domain wall\ncollapses rapidly after its formation. As shown in the simulations\nthis assumption is quite reasonable for walls whose size is much\nsmaller than the horizon. However, a large wall does not shrink until\nits size becomes smaller than the horizon, which leads to the increase\nof the density of axionic domain wall and as a result the density of\nthe axion produced by the collapse of the wall. Therefore\neq.(\\ref{omega-axion2}) may underestimate the actual axion density and\nthe axion from axionic walls may be more important than the cold axion.\n\nAnother important source of axions is an oscillating axionic string.\nLet us compare our result with the axion emission from the axionic\nstrings. The density of the axion from the strings is given by\n\\begin{equation}\n \\Omega_a({\\rm string}) \\simeq (1 - 0.01) (F_a\/10^{10}\\mbox{GeV})^{1.18},\n \\label{string}\n\\end{equation}\nwhere uncertainty of a factor of 100 is due to two different\nestimations by Davis\\cite{Davis} and Harari and Sikivie\\cite{Hagmann}.\nThus the importance of the axion from the wall depends on which\nestimation is correct. The difference comes from the different\nassumption for the energy spectrum of the emitted axion and it is hard to\njudge which assumption is better. In the case of the axionic domain\nwall the spectrum or average energy of emitted axions can be estimated\nby numerical simulations more easily since axions are mostly produced\nat the final stage of the collapsing wall whose size is much smaller\nthan the cosmic scale.\n\nIn conclusion, the relic density of the axion produced by the collapse\nof axionic walls is, at least, larger than the baryon density for $F_a\n\\gtrsim 10^{10}$ GeV and accounts for a part or all of the dark matter in\nthe universe. The axion from the axionic domain wall is more important\nthan the cold axion and comparable to that from strings if Harari and\nSikivie's estimation is correct. To make a more precise prediction of\nthe relic density we have to know the numeric parameter $\\alpha$ and\n$\\beta$, which is beyond the scope of the present work and will be\nstudied in future work.\n\n\\vskip 0.5cm\nMN is grateful to Professor K. Sato for his continuous encouragement.\nThis work was in part supported by the Japanese Grant in Aid for\nScience Research Fund of the Ministry of Education, Science and\nCulture (No. 3253). Numerical computations were partially performed by\nSUN SPARC stations at Uji Research Center, Yukawa Institute for\nTheoretical Physics, Kyoto University.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}