{"text":"\\section{Introduction} \\label{sec:intro}\n\nIt has been known for some time from ground-based observations that the outer solar atmosphere consisted of a photosphere, chromosphere and an extended corona, visible in eclipse observations.  With the identification of the coronal iron lines \\citep{edlen_1943}, the temperature of the corona was appreciated to be quite hot, on the order of 10$^6$ K.  \n\n\nAccess to space made it possible to observe the Sun from above the Earth's atmosphere which absorbs the ultraviolet (UV), extreme-ultraviolet (EUV) and X-ray regions of the spectrum.  These latter spectral regions contain continua and spectral lines formed over a broad range of temperatures and provide a detailed picture of the solar atmosphere.  The first ultraviolet spectra were obtained from a V-2 rocket \\citep{baum_tousey_1946}.  The early measurements have been summarized by \\citet{tousey_1963} and \\citet{pottasch}.  In the paper by Pottasch, he goes on to present an analysis of the intensities from a variety of observations spanning the wavelength range from 100 to 1900 \\AA.  The primary goal of the Pottasch analysis was to determine the relative abundances of the elements.  In doing this, the emission measure of each ion is determined at the temperature that maximizes the contribution function $G(T)$ (see Equ. \\ref{equ:intensity} - \\ref{equ:goft}).  These values are multiplied by the relative abundances that provide as little scatter as possible in a plot of the various emission measures as a function of temperature.  His emission measure plot reveals a curve that will become quite familiar, with the minimum of the emission measure occurring near 2 $\\times$ 10$^5$ K,  increasing toward chromospheric temperatures and increasing toward coronal temperatures.    \\citet{jordan_abund_emd} performed a similar analysis with the XUV spectra reported by \\citet{hall_xuv}.  The spectra contained emission lines formed at higher temperatures and the emission measures peaked at a temperature of 1.4 $\\times$ 10$^6$ K.\n\nThe idea that there exists a continuous differential emission measure (DEM) came to be used in the analysis of solar spectra.  \\citet{batstone_1970} used a a DEM consisting of 4 steps to explain their X-ray observations.  \\citet{chambe_1971} presented a continuous DEM of an active region that peaked between 1 and 2 $\\times$ 10$^6$ K and fell off exponentially at higher temperatures.  \\citet{walker_1974} analyzed whole-sun X-ray spectra in terms of a DEM and found a shape similar to that of \\citet{chambe_1971}.  \\citet{batstone_1970}, \\citet{walker_1974} and various other authors have employed either least-squared fits or chi-squared minimization techniques to determine their model parameters.  In the case of \\citet{chambe_1971}, the exact process is not clear.\n\nAfter these initial DEM analyses, it became clear that the derivation of a detailed, smooth DEM had problems.  These were first pointed out by \\citet{craig_brown}.  In order to overcome these difficulties, Bayesian techniques have been used to infer the DEM from spectral line observations.  \\citet{kashyap_drake} developed a Monte-Carlo-Markov-Chain (MCMC) code with a Metropolis sampler in the Interactive Data Language (IDL).  \\citet{warren_bayes} employed a sparse Bayesian analysis to examine the DEM that can be inferred from synthetic spectral line intensities. \\citet{hannah_kontar} developed a regularized inversion technique to determine the DEM from observed intensities.  All of these methods apply a smoothing or regularization to the derived DEM.  Here, a different approach is taken to understand the limitations of observed sets of solar emission lines to determine or infer the emission measure structure of the solar atmosphere by means of minimal models requiring few parameters.  No smoothing is applied and the only constraint, other than that provided by the observations, is that the emission measures are positive-definite.\n\n\\section{The approach} \\label{sec:approach}\n\nThe determination of emission measures in the solar atmosphere is based on a number of assumptions about the physical nature of the corona.  They have been stated previously by \\citet{kashyap_drake} in their Section 3.1 and essentially repeated here.  The source region is optically-thin for the EUV lines analyzed here, the ionization equilibrium is determined by a balance between collisional ionization and recombination, the ions are excited by collisional excitation by electrons and protons with a Maxwell-Boltzmann distribution, and the elemental abundances do not vary within the source region.  In addition, it is assumed that there is an unknown coronal heating mechanism maintaining the hot corona.\n\nIn previous analyses it has generally been assumed that the differential emission measure (DEM) is a continuous, smoothly varying function of temperature.  The approach taken here is to assume that the intensities of the observed spectral line can be reproduced by a set of isothermal emission measures.  \\citet{feldman_sumer_isot} and \\citet{landi_isot} have demonstrated that for some spectra observed above the limb, the  intensities can be reproduced by a single temperature-emission measure pair (T-EM) , where EM is given by the line of sight integral \\mbox{$\\int$ N$_e$ N$_H$ d{\\it l}}.\n\nTwo sets of spectral line observations of \\citet{brosius} are examined here.  These observations include a region of the quiet Sun and an active region.  They will be discussed further in Sections \\ref{sec:qs} and \\ref{sec:ar}.\n\nFor each line in the spectrum, the {\\it G(T)} function is calculated for a range of temperatures using the atomic parameters in the CHIANTI atomic database \\citep{dere_v1, delzanna_v10} with the ChiantiPy software package \\citep{ChiantiPy}.  The photospheric elemental abundances of \\citet{asplund_abund} as updated by \\citet{scott_abund_2, scott_abund_1} are used.  All CHIANTI lines produced by elements having an abundance greater than 10$^{-7}$ that of hydrogen and within 0.05 \\AA\\ of the observed wavelength are included.  In very few cases is there significant blending.  The temperatures are calculated on a grid with exponential increments of 10$^{0.005}$ K over the temperature range 2.5$\\times$10$^5$ to 4.5$\\times$10$^6$ for the quiet Sun and 2.5$\\times$10$^5$ to 5.6$\\times$10$^6$ for the active region.  An average value for the electron density is determined from the available density sensitive lines for each region.  Thus, for a prescribed set of temperatures and emission measures, a spectrum can be calculated.  The question is to find which set of temperatures and emission measures provide the greatest likelihood that the weighted deviations of the observations from the predictions conform to a normal distribution.\n\nA Bayesian approach is followed through the use of a Monte-Carlo-Markov-Chain (MCMC) process to determine the most likely set of temperature-emission (T-EM) pairs that reproduce the observed line intensities.  \\citet{kashyap_drake} have provided a good summary of the MCMC technique coupled with the use of the Metropolis sampler \\citep{metropolis}.  Here, the MCMC modeling is performed by the open-source Python PyMC3 package \\citep{pymc}.  It includes both the Metropolis sampler and the No U-Turn Sampler (NUTS) \\citep{NUTS}.  The Metropolis sampler is one of the first that was developed and can be somewhat inefficient since all proposed steps do not lead to a greater likelihood and the proposal process must be repeated, effectively taking a \"U-turn\" while traversing the Markov chain.  The NUTS sampler was developed for the case of continuous parameter distributions to provide a better and faster sampling algorithm.   The authors list a number of improvements over other samplers, for instance, the ability to efficiently sample with \"minimum human interaction.\"   The model errors are treated as a normal distribution with a mean of zero and a standard distribution as discussed.  The likelihood that is considered by the MCMC chain is the likelihood that the errors are consistent with the assumed normal distribution.  It is necessary to calculate the contribution functions on a grid before doing the MCMC sampling.  With in the framework of the PyMC3 package, it is not possible to recalculate them for each sampling iteration.  At each step, the sampling proposes a set of indices with which the contribution functions can be immediately determined.\n\n\nThe intensity of a spectral line produced by a  single ion is given by \n\\begin{equation} \nI\\,(i \\to f) \\, = \\,  \\frac{h \\nu}{4\\pi} \\, A(i \\to f) \\, \\int N_i \\, d{\\it l}\n\\label{equ:intensity}\n\\end{equation} \nwhere $h\\nu$ is the energy of the emitted photon,  $A(i\\to f)$ is the radiative decay rate from initial level {\\it i} to final level {\\it f}, and $N_i$ is the population density of level {\\it i} that is integrated along the line-of-sight {\\it l}.  If the energy $h\\nu$ is given in erg, then the units of $I(i\\to f)$ are in erg~cm$^{-2}$~s$^{-1}$~sr$^{-1}$.\n\nThe values of $N_i$ are obtained by solving the equilibrium level balance equations that model the atomic processes populating and de-populating the levels. The atomic parameters include radiative decay rates, electron and proton collisional rate coefficients, autoionization rates, and level-resolved recombination rates from the next-higher ionization state.  The solution of these equations provides the fraction of the ion X$^{+q}$ of the element X in level i.  In the following equation, this is denoted as N$_i$($\\rm{X}^{+q}$)\/N($\\rm{X}^{+q}$).  Once these value are obtained, it is necessary to calculate the value of N$_i$.  This is commonly done by the multiplication of a series of ratios that can be determined.\n\n\\begin{equation}\nN_i = \\frac{N_i(\\rm{X}^{+q})}{N(\\rm{X}^{+q})} \\, \\frac{N(\\rm{X}^{+q})}{N(\\rm{X})} \\, \\frac{N(\\rm{X})}{N_{\\rm{H}}} \\, N_{\\rm{H}}\n\\end{equation}\n\nHere, N($\\rm{X}^{+q}$)\/N($\\rm{X}$) is the fraction of the ion X$^{+q}$ relative to the element X and is obtained from calculations of the statistical equilibrium between ionization and recombination rates.  N($\\rm{X}$)\/N$_{\\rm{H}}$ is the abundance of the element X relative to hydrogen and is taken from compilations of a variety of measurements.  In very many cases, N$_i$($\\rm{X}^{+q}$)\/N($\\rm{X}^{+q}$) scales linearly with the electron density N$_e$ and it is convenient to divide this term by N$_e$ and then multiply the whole expression by N$_e$.  Then, we arrive at an expression for the predicted line intensity\n\n\\begin{equation} \nI\\,(i \\to f) \\, = \\, \\frac{h \\nu}{4\\pi} \\, A(i \\to f) \\, \\frac{N_i(\\rm{X}^{+q})}{N_e \\, N(\\rm{X}^{+q})} \\, \\frac{N(\\rm{X}^{+q})}{N(\\rm{X})} \\, \\frac{N(\\rm{X})}{N_{\\rm{H}}} \\, \\int  N_e N_{\\rm{H}} \\, d{\\it l}\n\\end{equation}\n\nThe G(N$_e$, T) is defined as\n\n\\begin{equation}\nG(N_e, T) \\, = \\, \\frac{h \\nu}{4\\pi} \\, A(i \\to f) \\, \\frac{N_i(\\rm{X}^{+q})}{N_e \\, N(\\rm{X}^{+q})} \\, \\frac{N(\\rm{X}^{+q})}{N(\\rm{X})} \\, \\frac{N(\\rm{X})}{N_{\\rm{H}}}\n\\end{equation}\n\nand \n\n\\begin{equation}\nI\\,(i \\to f) \\, = \\, G(N_e, T) \\, \\int N_e N_{\\rm{H}} \\, d{\\it l}\n\\label{equ:goft}\n\\end{equation}\n\nwhere the line-of-sight emission measure is $\\int N_e N_{\\rm{H}} \\, d{\\it l}$.\n\n\nFor the present analyses, the emission measure distribution is given by\n\n\\begin{equation}\nEM(T) = \\sum_i \\delta(T - T_i) EM_i \n\\end{equation}\n\n\n\nFor each T-EM pair, there are two variables, the index for the temperature array and the EM.  The temperature indices i$_n$ are constrained such that\n\\begin{equation}\ni_{min} \\leq i_0 < i_1 < i_2 \\cdots i_m \\leq i_{max} \n\\end{equation}\nwhere i$_{min}$ is the minimum index for the temperature array, usually zero, and i$_n$ are the temperature indices where n = 0 through the number of temperatures less one.  The maximum index i$_{max}$  is the largest index value of the temperature array.  The temperature indices are sampled with a Metropolis sampler available in PyMC3 and are modeled as a discrete-uniform distribution provided by PyMC3.  The EM values are constrained to be positive-definite by the use of the logarithm of the EM as the sampling variable.  Each value of EM$_i$ is modeled as a continuous-uniform distribution, also provided by PyMC3, where minimum and maximum values are set.  The values of EM$_i$ are sampled with the NUTS sampler \\citep{NUTS}.  The uniform prior distributions are relatively uninformed but maximum and minimum values can be set for the temperatures and each EM$_i$ and a starting point for the sampling provided.\n\n\nThe model consists of a prescribed number T-EM pairs.  Sequential models are constructed, beginning with 2 pairs.  Initial values of the T-EM are estimated from the {\\it em-loci} plots such as displayed in Figs. \\ref{fig:qs_emplot} and \\ref{fig:ar_emplot}.  After sufficient tuning steps, the MCMC samplers are run to provide a posterior distribution of the temperature and EM values that best reproduce the observed spectral intensities.  From the posterior distributions, the means are determined for each value of T and EM and inserted back into the model to predict the spectrum.  A value of $\\chi^2$ can then be calculated for each set of T-EM pairs.  \n\\begin{equation}\n\\chi^2 = \\sum_i ((I_i - P_i)\/\\sigma_i)^2\n\\label{equ:chi-squared}\n\\end{equation}\nwhere I$_i$ is the observed intensity for line i, P$_i$ is the predicted intensity and $\\sigma_i$ is an estimate of the combined error in observed and predicted intensities.  As in \\citet{dere_serts_densities}, an estimate of the standard deviation $\\sigma = w \\times I_i$ is used.  In \\citet{dere_serts_densities} a value of $w$ = 0.2 was determined for comparing lines of the same ion.  Since the appropriate value of $w$ is not known beforehand, it has been necessary to determine it in an iterative manner.  First, a 2 T-EM model with an estimate for $w$ was used and evaluated by means of MCMC modeling.  The values of the T-EM pairs are determined from the mean of the posterior distributions of each of the parameters.  These values are then used to predict the spectral line intensities and the standard deviation of the difference between the predictions and the observations. This value of the standard deviation determines a new value of $w$.  The 2 T-EM model is run again with this new value of $w$ to determine an improved estimate of $w$.  This process is then repeated with the 3, 4, and 5 T-EM models in that order.  A final estimate of $w$ = 0.3 is arrived at with the 4 and 5 T-EM models.  All models are then run with this final value of $w$ = 0.3.  In \\citet{dere_serts_densities} a value of $w$ = 0.2 was determined.  The difference can be explained by the fact that in \\citet{dere_serts_densities} only intensities of lines of the same ion were compared.  Consequently, whatever errors that are caused by uncertainties in the elemental abundances or ionization equilibrium are absent.  \\citet{brosius} lists uncertainties for their intensity measurements.  These are roughly half of the value of 0.3 used here that accounts for the combined uncertainties in the intensity measurements, the atomic data and the model.\n\nThe procedure is to iterate over the number of T-EM pairs, starting with 2 pairs and continuing to greater numbers of pairs, using the final value of $w$ = 0.3.  The results for each pair is compared by means of the reduced chi-squared \\citep{bevington}.\n\\begin{equation}\n\\chi^2_{\\nu} = \\chi^2 \\, \/ \\, \\nu\n\\label{equ:reduced_chisq}\n\\end{equation}\nwhere $\\nu$ is the degrees of freedom N$_{obs}$ - N$_{params}$, N$_{obs}$ is the number of observations and N$_{params}$ is the number of parameters in the model.  The number of parameters is 2 $\\times$ the number of T-EM pairs plus 1.  The fact that it has been necessary to determine the value of $w$ accounts for the additional parameter.  The value of N$_{obs}$ is taken as the number of observed spectral lines.  \nValues of $\\chi^2_{\\nu}$ should be about 1 if the data and the model are appropriate.  \"Values of $\\chi^2_{\\nu}$ much larger than 1 result from large deviations from the assumed distribution and may indicate poor measurements, incorrect assignment of uncertainties, or an incorrect choice of probability function.  Very small values of $\\chi^2_{\\nu}$ are equally unacceptable and may imply some misunderstanding of the experiment \\citep{bevington}\".\n\n\n\n\n\n\\section{The analysis of a quiet sun spectrum} \\label{sec:qs}\n\nThe observed spectral line intensities are taken from Table 2 of \\citet{brosius} and were obtained in a quiet region of the Sun in 1993. Spectral line observations were made in a wavelength range between 274 and 417 \\AA\\ and include 57 spectral lines formed by 19 ions.   The intensities tabulated by \\citet{brosius} were obtained by averaging over the 282 arc-sec slit. The quoted spatial resolution is about 5 arc-sec.     \n\nDensity sensitive line ratios in the data set have been analyzed by \\citet{dere_serts_densities}.  Line intensity ratios from ions of \\ion{Mg}{viii}, \\ion{Si}{ix}, \\ion{Si}{x}, \\ion{Fe}{xi}, \\ion{Fe}{xii},  \\ion{Fe}{xiii}, and \\ion{Fe}{xiv} were used to derive an average electron density of 5 $\\times$ 10$^{8}$ cm$^{-3}$.  Of these ions, the most robust results were obtained with \\ion{Si}{x} and \n\\ion{Fe}{xii}, \\ion{Fe}{xiii}, and \\ion{Fe}{xiv}.  The analysis used both a chi-squared minimization scheme and MCMC sampling of the electron density-EM space at a temperature specific to each ion.  There two methods provided a reasonable agreement with each other.\n\nMost of the quiet sun lines listed by \\citet{brosius} were included in the present analysis, with the exception of \\ion{He}{ii} $\\lambda$ 304.  It is formed at a much lower temperature than the other lines and would likely require its own T-EM pair that would be indeterminate.  In addition, one of the line identifications were updated.  The line at 359.374 \\AA\\ identified by \\citet{brosius} as \\ion{Ne}{v} (2s$^2$\\,2p$^2$ $^3$P$_2$  - 2s\\,2p$^3$ $^3$S$_1$) is found to be blended with a much stronger \\ion{Ca}{viii} line (3s$^2$\\,3p $^2$P$_{3\/2}$ - 3s$^2$\\,3d $^2$D$_{5\/2}$).  This is true both in the quiet Sun spectra and the active region spectra discussed later in Section \\ref{sec:ar}.  The final list consists of 53 spectral lines produced by 19 ions.  The temperature grid consisted of 271 exponentially spaced increments of 10$^{0.005}$ K from 2.5 $\\times$ 10$^5$ to 4.5 $\\times$ 10$^6$ K.  The {\\it G(T)} functions for each line, including possible blends, were calculated over this temperature range using the atomic data in Version 10 of the CHIANTI database \\citep{delzanna_v10}.\n\n\nFor the purposes of displaying the EM-{\\it loci} of this data set, an abbreviated spectral line set was created with only a single line per ion.  The selection process was very straightforward with the shortest wavelength line of each ion selected, basically, the first in the list.  These are shown in Fig. \\ref{fig:qs_emplot}.  Also, the final solution, a four T-EM pair model, is displayed as the solid dots in the figure.\n\n\\begin{figure}[ht!]\n\\plotone{tab2_1993_QS_v4_abbrev_emplot_results_points_pub.eps}\n\\caption{The EM-{\\it loci} for the abbreviated list of spectral lines in the 1993 quiet Sun spectra.  In addition, the values for the four T-EM pair final solution (dots).  \\label{fig:qs_emplot}}\n\\end{figure}\n\n\n\nOnce the appropriate standard deviation parameter $w$ has been determined, as discussed above in $\\S$ \\ref{sec:approach}, the determination of the most likely solutions proceeds.  By starting with a 2 T-EM pair model, it is first necessary to find prior estimates of the 2 temperatures and 2 emission-measures.  This is done by visually inspecting the EM-{\\it loci} plots displaying all of the EM-{\\it loci} curves, considerably more than in Fig. \\ref{fig:qs_emplot}, to find where there are many EM-{\\it {\\it loci}} curves that form a dense pattern, following \\citet{feldman_sumer_isot, landi_isot}.  In some cases the estimate of the prior parameters results in a posterior distribution that is located at a distance from the prior values.  However, a very poor choice of the prior starting point can lead to an unrealistic outcome.  Following the analysis with the 2 T-EM model a similar analysis is performed with the 3 T-EM model with improved information about the priors obtained from the 2 T-EM modeling.  This is followed by application of the 4 and 5 T-EM models.  For the larger number of T-EM pairs, the prior estimates for the temperature are often set at equally spaced values between reasonable minimum and maximum values.  These latter models provide the best estimates of the standard deviation of the observed intensities with respect to the predicted intensities.\n\nThe process of evaluating 2, 3, 4, and 5 T-EM pair models was then repeating but with a common value of $w$ = 0.3.  For each evaluation of the models, the best parameters are determined for each T-EM model pair.  The posterior distributions for the 4 T-EM model for the temperatures  are shown in Fig. \\ref{fig:qs_t_stackplot} and the posterior distributions for the EM are shown in Fig. \\ref{fig:qs_em_stackplot}.  From these posterior distributions, the values for the 4 T-EM pairs are derived as the mean and standard deviations.  These values are presented in Table \\ref{tab:qs_T-EM}.  The values of the standard deviation of the derived posterior parameters are quite small.  If they were to be plotted in Fig. \\ref{fig:qs_emplot}, they would be almost invisible.\n\n\n\nThese derived parameters, consisting of the mean value of the posterior distributions, are then used to calculate a predicted spectrum and the value of $\\chi^2$, following Equ. \\ref{equ:chi-squared}.  The values found for $\\chi^2$ are 110 for the 2 T-EM pairs model, 79 for the 3 T-EM pairs model, 47 for the 4 T-EM pairs model, and 46 for the 5 T-EM pairs model.  The values of $\\chi^2_{\\nu}$ (Equ. \\ref{equ:reduced_chisq}) are shown in Fig. \\ref{fig:qs_reduced_chisq} {\\it versus} the number of T-EM pairs.  From this figure it can be seen that using 5 T-EM pairs does not improve the fit over the use of 4 T-EM pairs.  In fact, the solution for the 5 T-EM pairs has one T-EM pair where, after sufficient tuning with the MCMC procedure, the value of the EM is so low that it does not effectively contribute to the predicted spectrum.\n\n\n\\begin{table}[ht!]\n\\begin{center}\n\\caption{ Temperature, Emission Measures and their standard deviations ($\\sigma$) for the 4 T-EM quiet Sun model}\n\\label{tab:qs_T-EM}\n\\begin{tabular}{cc}\n\\hline\nT $\\pm$ $\\sigma$ (10$^6$ K) & EM $\\pm$ $\\sigma$ (10$^{26}$ cm$^{-5}$) \\\\\n\\hline\n0.612  $\\pm$ 0.0018  &   1.21 $\\pm$ 0.26 \\\\\n1.08   $\\pm$ 0.036  &   2.99 $\\pm$ 0.36 \\\\\n1.62   $\\pm$ 0.024 &   10.7  $\\pm$ 0.36 \\\\\n2.33   $\\pm$ 0.045 &     6.37 $\\pm$ 0.32 \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\n\n\n\n\n\n\\begin{figure}[ht!]\n\\plotone{qs_4t_t_stack_prob_dens.eps}\n\\caption{The posterior probability density for the temperature of the 4 T-EM quiet Sun model.  \\label{fig:qs_t_stackplot}}\n\\end{figure}\n\n\\begin{figure}[ht!]\n\\plotone{qs_em_hist_stack_prob_dens.eps}\n\\caption{The posterior probability density for the EM of the 4 T-EM quiet Sun model.  \\label{fig:qs_em_stackplot}}\n\\end{figure}\n\n\\begin{figure}[ht!]\n\\plotone{tab2_1993_QS_reduced_chisq_vs_nT.eps}\n\\caption{The values of $\\chi^2_{\\nu}$ as a function of the number of T-EM pairs in each quiet Sun model.  \\label{fig:qs_reduced_chisq}}\n\\end{figure}\n\nIn Fig. \\ref{fig:qs_weighted_dev} the weighted deviations $(I_i - P_i)\/\\sigma_i$ together with the average and 1, 2 and 3 standard deviations (std) are shown.  The values are roughly consistent with a normal distribution with 77\\% being within 1 std, 92\\% within 2 std and 98\\% within 3 std, compared with the values for the normal distribution of 68\\%, 95\\% and 99.7\\%, respectively.\n\n\\begin{figure}[ht!]\n\\plotone{tab2_qs_diffOverWInt_wvl_mean_std.eps}\n\\caption{The values of the weighted deviations for the quiet Sun spectra {\\it vs} wavelength.} \\label{fig:qs_weighted_dev}\n\\end{figure}\n\n\n\\section{The analysis of an active region spectrum} \\label{sec:ar}\n\nThe observed spectral line intensities are taken from Table 1 of \\citet{brosius} and were obtained in a solar active region in 1993. Spectral line observations were made in a wavelength range between 274 and 417 \\AA\\ and include 65 spectral lines formed by 25 ions. The spectral lines are formed over a temperature range from 5 $\\times$ 10$^5$ to 1.6 $\\times$ 10$^7$ K, \\citep{brosius}. The intensities tabulated by \\citet{brosius} were obtained by averaging over the 282 arc-sec slit. The quoted spatial resolution is about 5 arc-sec.\n\n\n\\subsection{Electron densities in the active region} \\label{subsec:ne}\n\nThe electron densities in the active region have been derived by means of density-sensitive line-ratios of the ions \\ion{Fe}{xi}, \\ion{Fe}{xii},  \\ion{Fe}{xiii}, and \\ion{Fe}{xiv}.  The determination of the densities has been performed by a straightforward $\\chi^2$ minimization process as used by \\citet{dere_serts_densities}.  The observed intensities of each line of the diagnostic ion are divided by their contribution function {\\it G(T)} as a function of electron density at the temperature where the {\\it G(T)} function peaks.  For \\ion{Fe}{xiii} this temperature is 1.78 $\\times$ 10$^6$ K.  The EM-{\\it loci} for the \\ion{Fe}{xiii} lines are shown in Fig. \\ref{fig:ar_fe_13_emplot_best_68_95_r3.eps}.  In addition, the density that provides the best fit, as well as, the regions of 68\\% and 95\\% statistical confidence are also shown in Fig. \\ref{fig:ar_fe_13_chisq_vs_ne.eps}.  The regions of confidence are found by using the prescription of \\citet{lampton}.  This method was found to be comparable to a determination of these quantities through an MCMC analysis of the quiet Sun density diagnostics \\citep{dere_serts_densities}.\n\n\\begin{figure}[ht!]\n\\plotone{tab1_1993_AR_fe_13_emplot_best_68_95_r3.eps}\n\\caption{The EM-{\\it loci} for the \\ion{Fe}{xiii} lines for the active region used in the analysis {\\it vs} electron density.  The minimum of $\\chi^2$ occurs at an electron density of 1.4$\\times$ 10$^9$ cm$^{-3}$.  The regions for a statistical confidence of 68\\% and 95\\% are also shown. \\label{fig:ar_fe_13_emplot_best_68_95_r3.eps}}\n\\end{figure}\n\n\\begin{figure}[ht!]\n\\plotone{tab1_1993_AR_fe_13_chisq_68_95_r3.eps}\n\\caption{$\\chi^2$ as a function of electron density as a function of electron density for \\ion{Fe}{xiii} for the active region.  The minimum of $\\chi^2$ occurs at an electron density of 1.4$\\times$ 10$^9$ cm$^{-3}$.  \\label{fig:ar_fe_13_chisq_vs_ne.eps}}\n\\end{figure}\n\n\nThe same procedures used to derive electron densities from the \\ion{Fe}{xiii} lines have also been applied to the lines of \\ion{Fe}{xi}, \\ion{Fe}{xii}, and \\ion{Fe}{xiv}.  The results of this analysis are displayed in Fig. \\ref{fig:ar_densities.eps} where the best fit densities as well as the regions of 68\\% confidence are also shown.  For \\ion{Fe}{xi} only an upper limit can be obtained.  From these results, an average electron density of  2$\\times$10$^9$ cm$^{-3}$ is chosen.\n\n\\citet{brosius} found somewhat higher densities of 5 $\\times$ 10$^9$ cm$^{-3}$ in their 1993 active spectra.  As suggested by \\citet{dere_serts_densities}, the difference is largely due to the fact the only distorted wave calculations were available to provide the calibration of densities {\\it vs} line ratios.  These tend to underestimate some of the important excitation rates responsible for the dependence of the level populations on the electron density.\n\n\\begin{figure}[ht!]\n\\plotone{table1_1993_AR_densities_best_67.eps}\n\\caption{Electron densities derived from the lines of \\ion{Fe}{xi}, \\ion{Fe}{xii}, \\ion{Fe}{xiii}, and \\ion{Fe}{xiv} as a function of the temperature of their peak emissivity for the active region.  These are compatible with an average electron density of 2$\\times$10$^9$ cm$^{-3}$.  \\label{fig:ar_densities.eps}}\n\\end{figure}\n\n\n\\subsection{The emission measure distribution in the active region} \\label{subsec:ar-emd}\n\nThe temperature grid consists of 271 temperatures exponentially spaced by increments of 10$^{0.005}$ K from 2.5 $\\times$ 10$^5$ to 5.6 $\\times$ 10$^6$ K.  Again, not all of the 65 spectral lines in the \\citet{brosius} spectrum were used.  The final list consists of 63 spectral lines formed by 25 ions.  The emissivities of each line are calculated at an electron density of 2$\\times$10$^9$ cm$^{-3}$, otherwise, the procedures are the same as followed in Sec. \\ref{sec:qs} for the quiet Sun.  An abbreviated set of EM-{\\it loci} for the active region is displayed in Fig. \\ref{fig:ar_emplot}.  Also, the final solution for a four T-EM pair model, is displayed as the solid dots in the figure.\n\n\n\nFor the case of the best 4 T-EM pair model, the posterior probability densities for the temperatures are shown in Fig. \\ref{fig:ar_t_stackplot} and those for the emission-measures are shown in Fig. \\ref{fig:ar_em_stackplot}. The derived parameters, consisting of the mean value of the posterior distributions for temperature and emission measure, are then used to calculate a predicted spectrum and the value of $\\chi^2$ is calculated, following Equ. \\ref{equ:chi-squared} for each of the T-EM pair models.  The values found for $\\chi^2$ is 182 for the 2 T-EM pairs model, 91 for the 3 T-EM pairs model, 66 for the 4 T-EM pairs model, and 64 for the 5 T-EM pairs model.  The values of $\\chi^2_{\\nu}$ (Equ. \\ref{equ:reduced_chisq}) are shown in Fig. \\ref{fig:ar_reduced_chisq} with the number of observations N$_{obs}$ equal to the number of lines observed.  From this figure it can be seen that using 5 T-EM pairs does not improve the fit over the use of 4 T-EM pairs.  The solution for the 5 T-EM pairs has one T-EM pair that does not contribute much to the predicted spectrum.  The parameter values for the 4 T-EM model are provided in Table \\ref{tab:ar_T-EM}.  The values of the standard deviation are again quite small.\n\n\n\\begin{figure}[ht!]\n\\plotone{table1_1993_AR_v3_abbrev_emplot_results_points_pub.eps}\n\\caption{The EM-{\\it loci} for the abbreviated list of lines in the 1993 active region spectra.  In addition, the values for the four T-EM pair final solution (dots).  \\label{fig:ar_emplot}}\n\\end{figure}\n\n\n\\begin{figure}[ht!]\n\\plotone{tab1_ar_t_stacked_hist_v3.eps}\n\\caption{The posterior probability density for the temperature of the 4 T-EM active region model.  \\label{fig:ar_t_stackplot}}\n\\end{figure}\n\n\\begin{figure}[ht!]\n\\plotone{tab1_ar_em_prob_dens_stackplot.eps}\n\\caption{The posterior probability density for the EM of the 4 T-EM active region model.  \\label{fig:ar_em_stackplot}}\n\\end{figure}\n\n\\begin{figure}[ht!]\n\\plotone{table1_1993_AR_chisq_vs_nT.eps}\n\\caption{ The values of $\\chi^2_{\\nu}$ as a function of the number of T-EM pairs in each model of the active region. \\label{fig:ar_reduced_chisq}}\n\\end{figure}\n\n\n\\begin{table}[ht!]\n\\begin{center}\n\\caption{Temperature, Emission Measures and their standard deviations ($\\sigma$) for the 4 T-EM active region model}\n\\label{tab:ar_T-EM} \n\\begin{tabular}{cc}\n\\hline\nT $\\pm$ $\\sigma$ (10$^6$ K) & EM $\\pm$ $\\sigma$ (10$^{26}$ cm$^{-5}$) \\\\\n\\hline\n0.608  $\\pm$ 0.0068  &   4.34 $\\pm$ 0.31 \\\\\n1.08   $\\pm$ 0.0011  &   18.1 $\\pm$ 0.59 \\\\\n1.81   $\\pm$ 0.01 &   76.0  $\\pm$ 0.11 \\\\\n3.07   $\\pm$ 0.03 &    149. $\\pm$ 1.5 \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\n\nIn Fig. \\ref{fig:ar_weighted_dev} the weighted deviations $(I_i - P_i)\/\\sigma_i$ together with the average and 1, 2 and 3 standard deviations (std) shown for the 1993 active region spectra.  The values are roughly consistent with a normal distribution with 62\\% being within 1 std, 90\\% within 2 std and 100\\% within 3 std, compared with the values for the normal distribution of 68\\%, 95\\% and 99.7\\%, respectively.\n\n\\begin{figure}[ht!]\n\\plotone{ar_diffOverWghtInt_ave_std.eps}\n\\caption{The values of the weighted deviations for the active region spectra {\\it vs} wavelength.} \\label{fig:ar_weighted_dev}\n\\end{figure}\n\n\n\\section{Discussion} \\label{sec:discussion}\n\nThe DEM for these spectra have been previously determined by \\citet{brosius} and \\citet{kashyap_drake}.  In the \\citet{brosius} analysis, a cubic-spline with a small number of nodes is used fit the line intensities to provide a smooth reconstruction of the DEM.  \\citet{kashyap_drake} find a significant temperature grid of about 20-25 values (their Fig. 2) and provide 95\\% confidence limits by means of an MCMC process.  The two reconstructions are shown in Fig. 2 of \\citet{kashyap_drake}.  Perhaps the most noticeable differences between the two are the high values of the \\citet{brosius} DEM at 1 $\\times$ 10$^5$.  \\citet{brosius} state that the low temperature values of the DEM are constrained by upper limits or relatively uncertain measurements of \\ion{O}{iii}, \\ion{C}{iv} and \\ion{Mg}{v} lines but it is not stated where these line intensities come from and they are not listed in any of the tables.  The lowest temperature of importance as found by \\citet{kashyap_drake} and the EMs found in the present work are about 5-6 $\\times$ 10$^5$ K.  Otherwise, the DEMs of \\citet{brosius} and \\citet{kashyap_drake} at high temperature are mostly in agreement, taking into account the differences in approach.  The values of \\citet{brosius} are smooth, as expected, while those of \\citet{kashyap_drake} show some peaks in the reconstruction but the authors suggest that not all of these peaks are statistically significant. \n\nThe emission measure distributions derived here are particular to these data sets.  For example, the observed spectra do not contain any lines that would be formed at about 10$^5$ K in the transition region.  Consequently, the emission measure at those temperature can not be determined from this data.  Since these are disk observations, the transition region is most likely to be along the line of sight.  Similarly, nothing can be said about the existence of higher temperature plasmas above about 3 $\\times$ 10$^6$ K.  The observations are limited in their ability to constrain an emission measure distribution beyond a certain temperature range.\n  \n\\citet{craig_brown} pointed out, some time ago, that the inversion of spectral line intensities to determine a continuous DEM is very unstable.  Here, empirical models are used to represent the emission measure distribution and apply Bayesian inference techniques to determine the level of detail that can be recovered from the set of observed spectral line intensities.  What is found is that the emission measure distribution in the quiet and active Sun can be determined with a maximum of 9 quantities, 4 coupled T-EM pairs and the standard deviation of the data from the model, for the set of spectral line intensities analyzed here.  Models with higher numbers of T-EM pairs are not statistically relevant.  The set of measured line intensities do not provide sufficient constraints to derive a more detailed model.\n\n\\citet{kashyap_drake} reconstruct the DEM from the same spectra analyzed here, but, by requiring a smooth solution, they are able to determine roughly twice the number of emission measure values as found in the present analysis.  In addition, they define a temperature grid for which a large number of temperatures are included.  Consequently, the number of parameters that they derive are about 4 times that of the present analysis.  \\citet{hannah_kontar} employ a regularized inversion technique to determine the DEM of the active region core from EIS and SDO X-ray observations of \\citet{warren_2010_eis_ar}.  Their solution is also highly detailed.  \\citet{warren_bayes} employed a sparse Bayesian to examine the DEM that can be inferred from synthetic spectral line intensities created from a model DEM.  They derive 30 \"weights\" from 21 synthetic spectral lines and a broad-band EUV detector observation.  All three of these techniques use a smoothed or regularized solution.  This is clearly a very strong constraint on the DEM solutions.  \n\nThe use of smoothing and regularization comes from the idea that the DEM should be relatively smooth.  However, this is more of an arbitrary constraint and not one based on physical models or concepts.  If there are no physical models, then it is questionable as to what has been learned about the structure of the corona from the DEM.  The empirical models employed here only tell us what set of temperatures and emission measures are constrained by the observed spectra.  They offer only a very vague description of the solar atmosphere.  These techniques are best used in combination with physical models that make concrete predictions of the T-EM distribution.\n\nThe implication is that the derivation of a DEM or EM distribution is of little value, in and of itself.  However, the approach used here can result in some improvements in the determination of electron densities and relative elemental abundances.  In the case of determining electron densities from density-sensitive line ratios, it is necessary to select a temperature at which to calculate the predicted ratio.  This is often taken as the temperature that maximizes the {\\it G(T)} function.  \\citet{landi-L-function} have introduced the use of {\\it L-functions} as a method of reducing the uncertainties in the determination of the temperature for calculating the line ratios.  They separate the {\\it G(T)} function into two functions f$_{ij}$(N$_e$, T) and g(T) that multiply each other (see Equ. 9 of \\citet{landi-L-function}).  The function g(T) contains the temperature dependence that is common to all lines of a given ion.  The temperature to be used for the calibration of the line-ratio {\\it vs} density is found from the average of the temperature integrated over the product of g(T) and the DEM.  Under the current approach, there is no DEM.  In this case, one should construct a model using all of the T-EM pairs to derive the density, although this would be labor intensive.    \n\n\\citet{craig_brown} stated that one would only be able to derive a number of parameters that are much less than the number of spectral lines.  Their conclusion was based on the analysis of \\citet{herring_craig} where 2 T-EM pairs, or 4 parameters, were able to account for the X-ray emission in solar flares recorded in the 7 channels of the X-ray proportional counter on the Orbiting Solar Observatory 5.  For the quiet sun analysis here, the ratio is around 1 parameter for each of 6 spectral lines observed or 1 parameter for each of 2 ions that are observed.  For the active region analysis, the ratio is around 1 parameter for each of 7 spectral lines observed or 1 parameter for each of 3 ions that are observed.  If we take our ratio of the number of parameters to the number of ions observed, then these results generally agree with the previous conclusions of \\citet{craig_brown}.  They also pointed out the further constraints could be obtained from physical models but so far none have proved helpful.\n\n\n\n\n\\section{Conclusions}  \\label{sec:conclusions}\n\nA simple empirical model consisting of a specified number of discrete T-EM pairs has been used to explore the constraints that a set of observed spectral line intensities place on the derived emission measure distribution.  Using an MCMC process, it is determined that only 4 T-EM pairs are required to reproduce the spectral line intensities in both a quiet Sun region and an active region, for the set of observations analyzed here.  In addition, the errors in the derived parameters are determined from the MCMC posterior distributions.  In this case, they are relatively small.\n\nThe goal of this analysis has been to make an empirical determination of the ability of a set of emission line intensities to constrain the reconstruction of the emission measure distribution.  The conclusion to be drawn here is that any given set of emission line observations provide a limited amount of information with regard to the determination of the emission measure distribution.  Consequently, only a limited number of statistically meaningful parameters can be inferred, leading to reconstructions that are limited in terms of their information content, such as the temperature extent over which the distribution can be determined and the degree of detail that can be achieved.  This also would also apply to the determination of the differential emission measure.\n\n\\begin{acknowledgments}\n\nI thank Drs. E. Landi and P. Young for helpful comments on the manuscript.  I am grateful to Dr. Jeffrey Brosius for providing the tables from \\citet{brosius} in machine readable format.  This research has made use of NASA's Astrophysics Data System.  This work has been supported by NASA grants 80NSSC21K0110 and 80NSSC21K1785.\n\n\\end{acknowledgments}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
{"text":"\\section{Introduction}\nWhether at the microscopic or the cosmological scale, a major challenge in physics is understanding the real-time evolution of nonequilibrium quantum systems.  Classic examples of our limited knowledge in this area are hadronization of the quark-gluon plasma produced in heavy-ion collision and the expansion of the early universe.  While in principle these problems are amenable to numerical approaches upon classical computers, the exponentially large state space of quantum systems coupled with the numerical sign problem in both fermionic systems~\\cite{Troyer:2004ge} and real-time~\\cite{Alexandru:2016gsd} render such calculations intractable.\n\nThe promise of quantum computers is that the computational complexity of such problems can be reduced from exponential to polynomial.  This potential improvement is two-fold: one can represent the entanglement of quantum states directly and sign-problem free real-time calculations are possible. At present, we are restricted to fewer than 50 non-error-corrected qubits, which greatly restricts the class of problems we can attempt to simulate.  Despite these present limitations, calculations in systems of interest in nuclear physics~\\cite{Dumitrescu:2018njn,Roggero:2018hrn}, quantum field theory~\\cite{Klco:2018kyo}, condensed matter~\\cite{Macridin:2018gdw}, and quantum chemistry~\\cite{lanyon2010towards,PhysRevX.8.011021} have been achieved with as few as two qubits.  Typically, these calculations have relied upon hybrid algorithms that couple a few-qubit quantum computer solving a problem of exponentially bad classical computational complexity problem to a larger classical computer.\n\nIn this paradigm, we present in this paper the Evolving Density Matrices On Qubits (E$\\rho$OQ) algorithm, a hybrid quantum-classical technique for computing nonequilibrium dynamics of many-body quantum systems. In particular, we show how to compute the density matrix of a Hamiltonian $H_0$, with inverse temperature $\\beta$, and then evolve this mixed state in real-time by a different (potentially time-dependent) Hamiltonian $H_1$. The algorithm proceeds by computing on a \\emph{classical} computer a stochastic approximation to the density matrix $\\rho = e^{-\\beta H_0}$, via Density Matrix Quantum Monte Carlo~\\cite{PhysRevB.89.245124}. This approximate density matrix is passed to a quantum computer element-by-element, which performs time-evolution with a different Hamiltonian $H_1$, and then computes observables with the time-evolved density matrix $\\rho(t) = e^{-i H_1 t} \\rho e^{i H_1 t}$.\n\nPast theoretical work on computing thermal physics with a quantum computer has focused on performing the  thermal-state preparation on the quantum processor~\\cite{2016arXiv160907877B,2010PhRvL.105q0405B}. E$\\rho$OQ\\ differs from these approaches in allowing the computation of the thermal state to remain on the classical computer, using the quantum processor only for the classically intractable time-evolution.\n\nIn this work, we implement our algorithm for the 1D Heisenberg chain for $N\\leq5$.  The real-time evolution of this system has a long history of study on classical computers, starting with~\\cite{PhysRev.177.889}.  Since then, it has been used as a benchmark for developing time-dependent methods in quantum systems~\\cite{PhysRevE.71.036102,PhysRevB.77.064426,PhysRevLett.93.076401,1742-5468-2004-04-P04005}.\n\nIn Sec.~\\ref{sec:algorithm}, we describe the hybrid quantum-classical algorithm E$\\rho$OQ\\ in full detail.  Following this, a brief review of the 1D Heisenberg model is covered in Sec.~\\ref{sec:model}. Results using the Rigetti Forest, a quantum virtual machine (QVM)~\\cite{smith2016practical}, and Rigetti's 8-qubit quantum processor (QPU) 8Q-Agave, are presented in Sec.~\\ref{sec:results}, and conclusions are summarized in Sec.~\\ref{sec:discussion}.\n\n\\section{The Algorithm}\\label{sec:algorithm}\n\nThe first step of E$\\rho$OQ\\ produces a stochastic, sparse approximation to the density matrix using the Density Matrix Quantum Monte Carlo algorithm (DMQMC)~\\cite{PhysRevB.89.245124}, which we briefly summarize here. DMQMC is closely related to Diffusion Monte Carlo methods~\\cite{anderson1976quantum}, in which a population of `psips' explore the configuration space of a system through random walks in imaginary time $\\beta = i t$. Each psip is associated to a position basis state, and in the limit of large $\\beta$, the density of psips approximates the ground state wavefunction. In DMQMC, the psips explore the space of basis operators, and after evolution by a finite $\\beta$, the density of psips approximates the density matrix at inverse temperature $\\beta$.\n\nThe density matrix $\\rho(\\beta) = e^{-\\beta H}$ may be defined as the solution to the first-order differential equation\n\\begin{equation}\\label{eq:bloch}\n\t\\frac{\\d \\rho}{\\d \\beta} = - \\frac 1 2 \\left(H + H^\\dagger\\right) \\rho\\text,\n\\end{equation}\nwith the initial condition $\\rho(0) = 1$. DMQMC stochastically implements the first-order Euler difference approximation to \\eq{bloch}, with the density matrix represented by the collection of psips. To each psip is associated a basis operator $\\left|b_p\\right>\\left<a_p\\right|$ and a sign $\\chi_p$, determining the sign of the psip's contribution to the density matrix.  The approximate density matrix $\\tilde\\rho\\approx\\rho$ is given by a sum over all psips: the contribution to the density matrix of each psip $p$ is $\\chi_p \\left|b_p\\right>\\left<a_p\\right|$. Thus, $\\tilde\\rho$ is given by\n\\begin{equation}\\label{eq:rho}\n\t\\tilde\\rho = \\sum_p \\chi_p \\left|b_p\\right>\\left<a_p\\right|\\text.\n\\end{equation}\nThe algorithm begins by randomly placing psips along the diagonal of the density matrix, all with positive sign $\\chi=1$. This implements the desired initial condition for \\eq{bloch}.  The density matrix is then evolved in discrete steps of $\\Delta \\beta$, with $\\beta \/ \\Delta\\beta$ steps taken. At each step, every psip $p$ (living on site $\\left|b_p\\right>\\left<a_p\\right|$) performs four operations:\n\\begin{enumerate}\n\t\\item The psip may spawn a new psip on another site in the same column, $\\left|c\\right>\\left<a_p\\right|$ where $c \\ne b_p$, with probability $\\frac 1 2 \\left|\\left<c\\right|H\\left|b_p\\right>\\right|\\Delta\\beta$.\n\t\\item Similarly, the psip may spawn a new psip onto another site in the same row, $\\left|b_p\\right>\\left<c\\right|$ where $c \\ne a_p$, with probability $\\frac 1 2 \\left|\\left<a_p\\right|H\\left|c\\right>\\right|\\Delta\\beta$.\n\t\\item If $\\left<a_p\\right|H\\left|a_p\\right> + \\left<b_p\\right|H\\left|b_p\\right> > 0$, then the psip is removed from the simulation with probability $\\frac 1 2 \\left|\\left<a_p\\right|H\\left|a_p\\right> + \\left<b_p\\right|H\\left|b_p\\right>\\right| \\Delta \\beta$.\n\t\\item Alternatively, when $\\left<a_p\\right|H\\left|a_p\\right> + \\left<b_p\\right|H\\left|b_p\\right> < 0$, the psip is cloned, producing another psip on the same site. This occurs with probability $\\frac 1 2 \\left|\\left<a_p\\right|H\\left|a_p\\right> + \\left<b_p\\right|H\\left|b_p\\right>\\right| \\Delta \\beta$.\n\\end{enumerate}\n\nWhen the $\\beta \/ \\Delta\\beta$ executions of these four steps have completed, the resulting collection of psips gives an approximation to $\\rho(\\beta)$ via \\eq{rho}.\n\nWith the approximate density matrix $\\tilde\\rho$ determined, time-dependent expectation values are evaluated on a quantum processor.\nA time-dependent expectation value is given by\n\\begin{equation}\n\t\\left<\\mathcal O(t)\\right> = \\operatorname{{Tr}} \\mathcal O e^{-i H_1 t} \\rho e^{i H_1 t}\\text,\n\\end{equation}\nwhere $H_1$, the Hamiltonian used for time evolution, is distinct from the $H_0$ Hamiltonian used to define the density matrix.\nSubstituting the hermitized approximate density matrix $\\rho \\rightarrow \\frac 1 2 \\left(\\tilde \\rho + \\tilde\\rho^\\dagger\\right)$, we see that the expectation value may be approximated by a sum over psips:\n\\begin{align}\\label{eq:expval}\n\t&\\left<\\mathcal O(t)\\right> \\approx \\frac 1 {\\operatorname{{Tr}} \\tilde\\rho}\\times\\nonumber\\\\&\n\t\\sum_p \\operatorname{{Tr}} \\bigg(\\frac{1}{2} \\mathcal O e^{-i H_1 t}\n\t\\bigg[\n\t\t\\chi_p \\left|b_p\\right>\\left<a_p\\right|+\n\t\t\\bar\\chi_p \\left|a_p\\right>\\left<b_p\\right|\n\t\t\\bigg] e^{i H_1 t}\\bigg)\n\\end{align}\nFrom Eq.~\\ref{eq:expval} it can be seen that the decomposition of the density matrix into psips allows  one to time-evolve each psip independently as a pure state, avoiding the difficulty of constructing a mixed state on a quantum processor.\n\nPsips for which $a_p = b_p$ are termed `diagonal'. Expectation values $\\left<a_p\\right|\\mathcal O(t)\\left|a_p\\right>$ of diagonal psips may be evaluated straightforwardly on a quantum computer because they can be represented easily as a pure state.  In contrast, non-diagonal psips must be diagonalized before evaluation on a quantum processor. For real charges $\\chi_p$, a hermitized psip is diagonal in the basis $\\left|u_p\\right> = \\left|a_p\\right> + \\left|b_p\\right>$; $\\left|v_p\\right> = \\left|a_p\\right> - \\left|b_p\\right>$. Working in this basis (a different basis for each psip), the contribution to $\\langle \\mathcal{O}(t)\\rangle$ of the non-diagonal psips becomes\n\\begin{align} \\label{eq:expectation}\n\t&\\left<\\mathcal O(t)\\right> \\approx\\nonumber\\\\&\n\t\\frac 1 {\\operatorname{{Tr}} \\tilde\\rho}\n\t\\sum_p \\left[\n\t\t\\left<u\\right| e^{i H_1 t} \\mathcal O e^{-i H_1 t}\\left|u\\right>\n\t\t-\\left<v\\right| e^{i H_1 t} \\mathcal O e^{-i H_1 t}\\left|v\\right>\n\t\\right]\\text.\n\\end{align}\nIn this form, the expectation value is a sum of quantities each amenable to computation with a quantum computer. For a given set of psips specifying $\\tilde\\rho$, a separate instance of a general program is run on the quantum processor for each psip. Each program contains the same code for time-evolution and measurement, but a different sequence of operations for preparing the pure states. For non-diagonal psips, two programs must be executed, one for $\\left|u_p\\right>$ and one for $\\left|v_p\\right>$, while the diagonal psips require only one. Each program has the following steps:\n\\begin{enumerate}\n\t\\item Prepare the state $\\left|u_p\\right>$ (or $\\left|v_p\\right>$);\n\t\\item Time-evolve with $H_1$ for a fixed time $t$ via trotterization;\n\t\\item Measure $\\mathcal O$, and any other observables of interest simultaneously.\n\\end{enumerate}\n\nFor nearly all Hamiltonians of physical interest, the diagonal basis of the Hamiltonian is not efficiently accessible, and the time-evolution operator $e^{i H_1 t}$ must be approximated by trotterization. This is accomplished by decomposing the Hamiltonian into terms easily diagonalized: $H_1 = H_x + H_z$. The time-evolution operator is then $e^{i H_1 t} = \\left(e^{i H_x \\Delta t} e^{i H_z \\Delta t}\\right)^{t\/(\\Delta t)} + O(\\Delta t)$. In the case of \\eq{hamiltonian}, we trotterize with $H_x = -\\mu_x \\sum_i \\sigma^{(i)}_x$ and $H_z = -J_z \\sum_{<ij>} \\sigma^{(i)}_z \\sigma^{(j)}_z - \\mu_z \\sum_i \\sigma^{(i)}_z$.\n\nIn this paper, the observable of interest (transverse magnetization) may be measured by changing basis from the $Z-$ to the $X-$basis (a rotation of each qubit), and measuring all qubits simultaneously.\n\n\n\nOnce each psip has been evaluated by the quantum processor, the results are summed together (on the classical computer) via \\eq{expectation} to calculate the expectation value of the thermal state.\n\nThe efficiency of this algorithm is strongly influenced by the fact that the approximate density matrix $\\tilde\\rho$ may be extremely sparse, where the exact density matrix $\\rho$ is not. For an $N$-site system, the density matrix $\\rho$ has at least $2^N$ non-zero entries; we expect sufficiently accurate expectation values to be obtainable with a population of psips which scales only polynomially with $N$. Each psip corresponds to one or two calculations on the quantum computer; thus, the number of calculations required on the quantum computer is expected to be polynomial in $N$. \n\n\\section{The 1D Heisenberg Chain}\\label{sec:model}\nAs a demonstration of the algorithm, we simulate a 1D time-dependent Heisenberg spin chain with one coupling constant and two independent magnetic fields~\\cite{PhysRev.177.889,PhysRevE.71.036102,PhysRevB.77.064426,PhysRevLett.93.076401,1742-5468-2004-04-P04005}. The general Hamiltonian for this class of system is\n\\begin{equation}\\label{eq:hamiltonian}\n\tH(t) = -J_z(t)\\sum_{\\left<ij\\right>} \\sigma^{(i)}_z \\sigma^{(j)}_z - \\mu_x(t) \\sum_i \\sigma^{(i)}_x - \\mu_z(t) \\sum_i \\sigma^{(i)}_z\\text,\n\\end{equation}\nwhere $J_z(t)$ is the coupling constant between the $z-$axis aligned spin component of nearest neighbors, and $\\mu_x(t)$ and $\\mu_z(t)$ denote time-dependent magnetic fields aligned with the $x-$ and $z-$axes, respectively.  We take the spin chain to have periodic boundary conditions. In this paper, we will work in units where the inverse temperature is $\\beta = 1$, and restrict ourselves to a constant coupling $J_z(t)=1$ and longitudinal magnetic field $\\mu_z(t)$ which is $0$ for the $N=5$ system and $1$ for the $N=1$. The transverse magnetic field is permitted to be time-dependent.\n\nThe time-dependent observable we measure is the average transverse magnetization, given by\n\\begin{equation}\n{\\langle m_x(t)\\rangle}\\equiv \\frac 1 N \\sum_i\\sigma^{(i)}_x(t)\\text.\n\\end{equation}\nAs discussed in the previous section, this quantity is easily measured on the quantum processor.\n\n\\section{Results}\\label{sec:results}\nFor the purposes of this exploratory study, we compute ${\\langle m_x(t)\\rangle}$ for two cases: the $N=5$ spin chain on the Rigetti Forest QVM to empirically test the algorithm's correctness, and the single-spin case on the Rigetti 8Q-Agave quantum computer to study the sources of uncertainty arising in a physical quantum processor.\n\nWithout the additional sources of error inherent in a QPU, we are able to access larger systems on the QVM. We evolve the $N=5$ spin system with the Hamiltonian described by \\eq{hamiltonian} with $\\mu_x(t=0) = 1$ and $\\mu_x(t>0) = -1$. The longitudinal magnetic field is $\\mu_z = 0$.  For this calculation we use a trotterization time step of $\\Delta t = 0.1$. The imaginary time step was $\\Delta\\beta = 0.04$ for evolving the psips with $5000$ initial psips. Shown in \\fig{qvm} is ${\\langle m_x(t)\\rangle}$, in statistical agreement with the exact result.\n\n\\begin{figure}\n\t\\centering\n\t\\includegraphics[width=\\linewidth]{QVM}\n\t\\caption{The transverse magnetization ${\\langle m_x(t)\\rangle}$ for a $N=5$ site spin chain with coupling $J_z=1$, and an initial $\\mu_x(0)=1$ and $\\beta=1$, which is evolved with $\\mu_x(t> 0) = -1$. Results from the Forest QVM are shown by red circles and the exact result is denoted by the solid black line.\\label{fig:qvm}}\n\\end{figure}\n\nWhen run on an ideal quantum processor, as simulated by Rigetti Forest, E$\\rho$OQ\\ has two sources of uncertainty, both statistical: the approximation of $\\rho$ by a finite number of psips, and the intrinsic measurement noise on the quantum processor.  These sources of error are easily accounted for with standard methods such as bootstrapping as we do in this work.  Note, though, that the errors are correlated since the same set of psips (i.e., the same approximation to the density matrix) is used for all values of $t$.\n\n\\begin{figure}\n\t\\centering\n\t\\includegraphics[width=\\linewidth]{QPU}\n\t\\caption{The rescaled (see text) transverse magnetization ${\\langle m_x(t)\\rangle}\/\\langle m_x(0)\\rangle$ for a single spin, with initial $\\mu_x(0)=\\mu_z(0)=1$ and $\\beta=1.0$, which is evolved with $\\mu_x(t> 0) = -1$.  The results from Rigetti's 8Q-Agave QPU are shown in red circles while the exact result is denoted by the solid black line.\\label{fig:qpu}}\n\\end{figure}\n\nWe use the 8-qubit quantum processor 8Q-Agave to simulate a single spin, thermalized in a transverse magnetic field $\\mu_x(t=0)=1$, and time-evolved in a flipped magnetic field $\\mu_x(t=0)=-1$.  The longitudinal magnetic field is taken to be constant: $\\mu_z = 1$. For this calculation we use a trotterization time step of $\\Delta t = 0.2$. The imaginary time step was $\\Delta\\beta = 0.04$, with $1000$ initial psips. The results of this execution of the algorithm are presented in \\fig{qpu}, again in good agreement with the exact result.\n\nThe physical 8Q-Agave, unlike the simulated Forest, is not an ideal quantum processor, and has several additional sources of error that must be accounted for.  Most prominently, measurements have so-called {\\em readout noise}. When measuring a qubit, there is some probability that the opposite state will be read instead.   If one assumes this readout noise is symmetric between the two states and independent of the gates used before a measurement is taken (empirically the case at our level of precision), this reduces the measured magnitude of ${\\langle m_x(t)\\rangle}$ by a constant factor, which can be corrected for by rescaling.  In \\fig{qpu}, we rescale ${\\langle m_x(t)\\rangle}$ by $\\langle m(0)\\rangle$, which appears to sufficiently remove the effect of readout noise. \n\nOther sources of error, more difficult to correct for, are also present. For instance, when a parameterized gate (such as a 1-qubit phase gate) is requested with angle $\\theta$, the actual gate implemented may have angle $\\theta + \\epsilon(\\theta)$, producing a systematic bias in all results using that value of $\\theta$. This and other unanticipated sources of systematic error may be accounted for by performing a calibration run with a simpler Hamiltonian (diagonal in the computational basis). For this work we use $H'_1 = - \\mu_z \\sigma_z$: the error bars estimated for \\fig{qpu} are the quadrature average of the difference between the simulated results for $H'_1$ and the exact answer.\n\n\\section{Discussion and prospects}\\label{sec:discussion}\nIn this work, we have presented E$\\rho$OQ\\, a hybrid classical\/quantum algorithm for simulating out-of-equilibrium dynamics of thermal quantum systems, applying it to a simple system on both a quantum virtual machine and a quantum processor. E$\\rho$OQ\\ first computes an approximation of the density matrix upon a classical computer, evading the need to compute thermal physics or prepare a mixed state on a quantum computer.  The density matrix is then passed to a quantum processor to compute the time-evolution, thus avoiding the sign problem associated with real-time calculations on a classical computer.\n\nGoing forward, this algorithm could be applied to problems of greater physical interest. While the hadronization of the quark-gluon plasma or reheating in the early universe will require larger quantum processors than exist at present, the non-linear response of low-dimensional systems like spin chains and graphene as well as the response of a thermal neutron gas to neutrino scattering should be possible on near-future resources.  In order to do this, a better characterization of the errors present on today's physical quantum computers will be necessary --- a general concern for all quantum algorithms.\n\n\\begin{acknowledgments}\nH.L. and S.L. are supported by the U.S. Department of Energy under Contract No.~DE-FG02-93ER-40762.  The authors would further like to thank Rigetti for their assistance and access to their resources, Forest and 8Q-Agave.\n\\end{acknowledgments}\n\\bibliographystyle{apsrev4-1}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
{"text":"\\section{Canted Order Initial state}\nIn Ref.\\citep{Wichterich}, the N\\'eel state was considered because it is the natural ground state for $\\Delta \\rightarrow \\infty$. However, here we are interested in the theoretical (fundamental) problem of how much entanglement between distant spins can be generated from separable states and a global switch of couplings. The whole class of separable states is enormous as well as the class of product states of several spins. Thus we examine here the canted order pure initial state\n\\begin{equation}\n\\left|\\psi\\right\\rangle=\\mathop{\\otimes}_{k=1}^{N} \\left|\\psi_k\\right\\rangle=\\mathop{\\otimes}_{k=1}^{N} \\left( \\cos{\\frac{\\theta_{k}}{2}} \\left|\\uparrow_{k}\\right\\rangle + \\sin{\\frac{\\theta_{k}}{2}} \\left|\\downarrow_{k}\\right\\rangle \\right) \\, ,\n\\label{CantedState}\n\\end{equation}\nwhere $\\theta_k=(k-1) \\alpha$ and $0\\leq\\alpha\\leq 2\\pi$. This initial state may be prepared in several ways. The most obvious way is to have an {\\em uncoupled} Hamiltonian and a magnetic field varying in direction from site to site which tilts spins accordingly. Even with a magnetic field varying only in one (say the $x$) direction, one can prepare the above state for $\\alpha \\leq \\frac{\\pi}{2(N-1)}$ as the ground state of the coupled Hamiltonian\n\\begin{equation}\nH_{\\text{canted}}=-\\sum_{k=1}^{N-1} \\sigma_k^z \\otimes \\sigma_{k+1}^z + \\sum_{k=1}^N h_k^x \\sigma_k^x -h_1^z \\sigma_1^z \\, .\n\\end{equation}\nWith $h_k^x$ increasing with the site number, the ground state is a chain in which each spin is tilted counter-clockwise with an angle $\\alpha$ compared with the left neighboring spin (Fig.~\\ref{CantedState}). After N\\'eel order, a canted order is a next level of generalization that we can make for an initial product state of a spin chain. \n\n\\section{The Quench and the Generation of Entanglement}\n\\label{canted}\n We are motivated by the scope of solving for the entanglement of the distant end spins analytically for the given initial state.\nIt is with this view of making analytic progress that we let the system with the canted order initial state evolve with the $XX$ Hamiltonian. One can regard this as an instantaneous change (or a quench) from an uncoupled Hamiltonian with local magnetic fields or the Hamiltonian $H_{\\text{canted}}$ to the $XX$ Hamiltonian. Also note that in Ref.\\citep{Wichterich} it was found that $H_{\\text{XXZ}}$ with $\\Delta \\not= 0$ is worse in terms of generating entanglement in comparison to the case of $\\Delta=0$. This can be regarded as another motivation for evolving the system with $XX$ Hamiltonian. So one first prepares a canted order initial state of a few spins by one of the means described in the previous section and then, at time $t=0$ suddenly switches on a $XX$ Hamiltonian interaction between them. We are interested in the entanglement produced between spins $1$ and $N$ as a function of time due to this $XX$ Hamiltonian, $H$. \n\\begin{figure}[htbp]\n \\centering \n\\includegraphics[width=7cm]{Fig1.pdf}\n   \\caption{{\\protect\\footnotesize This figure shows the canted order state considered as an initial state (Eq.~\\ref{CantedState}).}}\n\\end{figure}\\\\\nTo make progress, we first use the Jordan-Wigner transformation,\n\\begin{equation}\n\\hat{c}_{l}^{\\dagger}=\\hat{S}_{1,l-1} \\hat{\\sigma}_l^{+}, \\quad \\hat{c}_{l}= \\hat{S}_{1,l-1} \\hat{\\sigma}_l^{-}\n\\end{equation}\nwith $\\sigma^{\\pm}=\\frac{1}{2}(\\sigma^{x}\\pm\\sigma^{y})$ and $\\hat{S}_{l,m}=\\mathop{\\otimes}_{k=l}^{m} (- \\hat{\\sigma}_k^{z})$,\nto write the Hamiltonian in terms of fermionic operators and then diagonalize $H$ to get the fermionic operators in the Heisenberg picture\n\\begin{equation}\n\\hat{c}_{k}(t)=\\sum_{l} f_{k,l} \\hat{c}_{l}(0), \\quad \\hat{c}_{k}^{\\dagger}(t)=\\sum_{l} f_{k,l}^{*} \\hat{c}_{l}^{\\dagger}(0) \\, ,\n\\end{equation}\n\\begin{equation}\nf_{k,l}(t)= \\frac{2}{N+1}\\sum_{m=1}^{N} \\sin(q_m k) \\sin(q_m l) e^{-i \\epsilon_m t} \\, ,\n\\end{equation}\nwith $\\epsilon_m=2 J \\cos(q_m)$ and $q_m=\\frac{\\pi m}{N+1}$.\nTo study the entanglement generated between the distant ends of the chain ($1, N$) we find the reduced density matrix in the basis $\\left\\{\\left| \\uparrow\\uparrow\\right\\rangle, \\left| \\uparrow\\downarrow\\right\\rangle,\\left| \\downarrow\\uparrow\\right\\rangle,\\left| \\downarrow\\downarrow\\right\\rangle\\right\\}$\n\\begin{equation}\n\\hat{\\rho}_{1,N}(t)=\\left(\\begin{array}{cccc}\n\\left\\langle \\hat{P}_{1}^{\\uparrow} \\hat{P}_{N}^{\\uparrow} \\right\\rangle & \\left\\langle \\hat{P}_{1}^{\\uparrow} \\hat{\\sigma}_{N}^{-} \\right\\rangle & \\left\\langle \\hat{\\sigma}_{1}^{-} \\hat{P}_{N}^{\\uparrow} \\right\\rangle & \\left\\langle \\hat{\\sigma}_{1}^{-} \\hat{\\sigma}_{N}^{-} \\right\\rangle\\\\\n\\left\\langle \\hat{P}_{1}^{\\uparrow} \\hat{\\sigma}_{N}^{+} \\right\\rangle & \\left\\langle \\hat{P}_{1}^{\\uparrow} \\hat{P}_{N}^{\\downarrow} \\right\\rangle & \\left\\langle \\hat{\\sigma}_{1}^{-} \\hat{\\sigma}_{N}^{+} \\right\\rangle & \\left\\langle \\hat{\\sigma}_{1}^{-} \\hat{P}_{N}^{\\downarrow} \\right\\rangle\\\\\n\\left\\langle \\hat{\\sigma}_{1}^{+} \\hat{P}_{N}^{\\uparrow} \\right\\rangle & \\left\\langle \\hat{\\sigma}_{1}^{+} \\hat{\\sigma}_{N}^{-} \\right\\rangle & \\left\\langle \\hat{P}_{1}^{\\downarrow} \\hat{P}_{N}^{\\uparrow} \\right\\rangle & \\left\\langle \\hat{P}_{1}^{\\downarrow} \\hat{\\sigma}_{N}^{-} \\right\\rangle\\\\\n\\left\\langle \\hat{\\sigma}_{1}^{+} \\hat{\\sigma}_{N}^{+} \\right\\rangle & \\left\\langle \\hat{\\sigma}_{1}^{+} \\hat{P}_{N}^{\\downarrow} \\right\\rangle & \\left\\langle \\hat{P}_{1}^{\\downarrow} \\hat{\\sigma}_{N}^{+} \\right\\rangle & \\left\\langle \\hat{P}_{1}^{\\downarrow} \\hat{P}_{N}^{\\downarrow} \\right\\rangle\n\\end{array}\\right) \\, ,\n\\label{rho1N}\n\\end{equation}\nwhere $\\hat{P}_{l}^{\\downarrow} = \\hat{\\sigma}_{l}^{-}\\hat{\\sigma}_{l}^{+},\\hat{P}_{l}^{\\uparrow} = \\hat{\\sigma}_{l}^{+}\\hat{\\sigma}_{l}^{-}$, $\\left\\langle ... \\right\\rangle=Tr(\\rho(t)...)$ and $\\rho(t)=e^{-i \\hat{H} t} \\rho(0) e^{i \\hat{H} t}$.\nWe will show the evaluation of one matrix element, say $(1,4)$, and the rest of the elements are found similarly. The matrix element $(1,4)$ is\n\\begin{equation}\n\\left\\langle \\hat{\\sigma}_{1}^{-} \\hat{\\sigma}_{N}^{-} \\right\\rangle= \\left\\langle \\hat{c}_1 (t) \\hat{c}_N (t) \\hat{S}_{1,N-1}\\right\\rangle= -\\left\\langle \\hat{c}_1(t) \\hat{c}_N (t) \\hat{S}_{1,N}\\right\\rangle \\, ,\n\\end{equation}\nwhere we used $\\hat{\\sigma}_{l}^{-} \\hat{S}_{l}=-\\hat{\\sigma}_{l}^{-}$. Hence\n\\begin{equation}\n\\left\\langle \\hat{\\sigma}_{1}^{-} \\hat{\\sigma}_{N}^{-} \\right\\rangle=-\\sum_{l,m} f_{1,l}(t) f_{N,m}(t) \\left\\langle \\hat{c}_l(0) \\hat{c}_m (0) \\hat{S}_{1,N} \\right\\rangle.\n\\end{equation}\nNow we have 3 cases, first if $l=m$, then\n\\begin{equation}\n\\left\\langle \\hat{c}_l(0) \\hat{c}_m (0) \\hat{S}_{1,N} \\right\\rangle=0.\n\\end{equation}\nSecond, If $l>m$, then\n\\begin{eqnarray}\n\\nonumber \\left\\langle \\hat{c}_l(0) \\hat{c}_m (0) \\hat{S}_{1,N} \\right\\rangle&=&\\left\\langle \\hat{S}_{1,l-1} \\hat{\\sigma}_{l}^{-}(0) \\hat{S}_{1,m-1} \\hat{\\sigma}_{m}^{-}(0) \\hat{S}_{1,N}\\right\\rangle\\quad\\\\\n\\nonumber &=&\\left\\langle \\psi_l \\right| \\hat{\\sigma}_{l}^{-}(0) \\left| \\psi_l\\right\\rangle \\left\\langle\\psi_m \\right|\\hat{\\sigma}_{m}^{-}(0) \\left| \\psi_m\\right\\rangle \\mathop{\\otimes}_{k=1}^{m-1}\\left\\langle\\psi_k \\right| -\\sigma^{z}_{k}\\left| \\psi_k\\right\\rangle \\mathop{\\otimes}_{k=l+1}^{N}\\left\\langle\\psi_k \\right| -\\sigma^{z}_{k}\\left| \\psi_k\\right\\rangle\\quad\\\\\n&=&\\frac{1}{4} \\sin \\theta_l \\sin \\theta_m \\mathop{\\otimes}_{k=1}^{m-1}(-\\cos \\theta_k) \\mathop{\\otimes}_{k=l+1}^{N}(-\\cos \\theta_k).\\quad\n\\end{eqnarray}\n\nwhere we used $(\\sigma^z_k)^2=1$. Finally, if $l<m$, then\n\\begin{eqnarray}\n\\nonumber \\left\\langle \\hat{c}_l(0) \\hat{c}_m (0) \\hat{S}_{1,N} \\right\\rangle&=&\\left\\langle \\hat{S}_{1,l-1} \\hat{\\sigma}_{l}^{-} (0) \\hat{S}_{1,m-1} \\hat{\\sigma}_{m}^{-} (0)  \\hat{S}_{1,N} \\right\\rangle\\quad\\\\\n\\nonumber &=&\\left\\langle \\psi_l \\right| \\hat{\\sigma}_{l}^{-}(0) \\left| \\psi_l\\right\\rangle \\left\\langle\\psi_m \\right|\\hat{\\sigma}_{m}^{-}(0) \\left| \\psi_m\\right\\rangle\\mathop{\\otimes}_{k=1}^{l-1}\\left\\langle\\psi_k \\right| -\\sigma^{z}_{k}\\left| \\psi_k\\right\\rangle\\mathop{\\otimes}_{k=m+1}^{N}\\left\\langle\\psi_k \\right| -\\sigma^{z}_{k}\\left| \\psi_k\\right\\rangle\\quad\\\\\n &=&\\frac{1}{4} \\sin \\theta_l \\sin \\theta_m \\mathop{\\otimes}_{k=1}^{l-1}(-\\cos \\theta_k)\\mathop{\\otimes}_{k=m+1}^{N}(-\\cos \\theta_k).\\quad\n\\label{analytic}\n\\end{eqnarray}\nSimilarly we can find analytic expressions for all the matrix elements of $\\rho_{1,N}$. However, these expressions, of the form of Eq.(\\ref{analytic}), have themselves to be evaluated numerically by virtue of the long products involved. This evaluation can be carried out efficiently even for very large values of $N$.\n\n\nAfter having determined the elements of $\\rho_{1,N}$, we are now in a position to compute the entanglement between spins $1$ and $N$ as a function of time. One way of quantifying the entanglement is to compute the concurrence, which is a measure of entanglement for two qubits \\citep{wootters}. However, note that this entanglement will, in the end, be used for teleporting arbitrary quantum states between two registers so that it is the ``fully entangled fraction\" rather than some measure of entanglement, which is the more relevant figure of merit. One would first apply the entanglement purification protocol in which a given number of copies of the state $\\rho_{1,N}$ can be converted, by local actions at both ends and classical communication, to a smaller number of maximally entangled states which can then be used for teleportation \\citep{Bennett2}. In order to test the possibility of the purification procedure we use the {\\em fully entangled fraction} defined as\n\\begin{equation}\nf=max(\\left\\langle e|\\rho|e \\right\\rangle) \\, ,\n\\end{equation}\nwhere the maximum is taken over all maximally entangled states $\\left\\{|e \\rangle\\right\\}$. The purification procedure is possible if $f>\\frac{1}{2}$ \\citep{Bennett2}. Moreover, if we were to use the state $\\rho_{1,N}$ itself for teleportation, then the fidelity of the teleported state is given by $F=\\frac{2f+1}{3}$ \\citep{Horodecki}. We will thus also compute the fully entangled fraction $f$ for $\\rho_{1,N}$ as a function of time and the canting angle $\\alpha$, as this is much more indicative of the eventual application in connecting quantum registers, namely teleportation. \\\\\nFigs~\\ref{CantedOrderN24} and \\ref{CantedOrderN50} show the dynamics of the fully entangled fraction and the concurrence of $\\rho_{1,N}$ as a function of $\\alpha$ for $N=24$ and $N=50$ spins respectively. Clearly, at around a scaled time very close to $t_{\\text{opt}}\\sim \\frac{N}{4J}$ (in fact the time is slightly larger than $t_{\\text{opt}}$) and for the N\\'eel state, a state $\\rho_{1,N}$ with the highest fully entangled fractions $f\\sim 0.78$ (for $N=24$) and $f\\sim 0.68$ (for $N=50$) is generated, which will not only directly allow one to teleport a state with a fidelities of $F\\sim 0.85$ (for $N=24$) and  $F\\sim 0.79$ (for $N=50$), but also enable entanglement purification (using many copies of $\\rho_{1,N}$) to obtain a state giving perfect teleportation fidelity. We focused here on the first emergence in time of long range entanglement. We note the sharp peak in the entanglement at $\\alpha=\\pi$ which corresponds to the N\\'eel state. This means that the proposed scheme generates entanglement for a N\\'eel state and not for the general canted order state. The narrowness of the peak in $\\alpha$, Fig.~\\ref{PeakWidth}, is especially striking implying essentially that unless a canted order state is {\\em extremely close} to the N\\'eel state it is ill-suited for generating any long distance entanglement. This uniqueness of the N\\'eel state among all the canted order states is one of our principal findings that adds to the knowledge with respect to Ref.\\citep{Wichterich}, where the high entanglement generating ability of only the N\\'eel state was noted. We will attempt to provide an explanation of this effect in the section on discussions in a later part of the paper.\n\n\\begin{figure}[htbp]\n\\centering \n\\includegraphics[width=14cm]{Fig2.pdf}\n   \\caption{{\\protect\\footnotesize (Color online) Dynamics of the concurrence and the fully entangled fraction, of spins $1$ and $N$ of the chain for $N=24$ as a function of the canting angle $\\alpha$. The scaled time $t$ is in units of $1\/J$.}}\n\\label{CantedOrderN24}\n\\end{figure}\n\n\\begin{figure}[htbp]\n\\centering \n\\includegraphics[width=14cm]{Fig3.pdf}\n   \\caption{{\\protect\\footnotesize (Color online) Dynamics of the concurrence and the fully entangled fraction, of spins $1$ and $N$ of the chain for $N=50$ as a function of the canting angle $\\alpha$. The scaled time $t$ is in units of $1\/J$.}}\n\\label{CantedOrderN50}\n\\end{figure}\n\n\\begin{figure}[htbp]\n\\centering \n{\\includegraphics[width=7cm]{Fig4.pdf}}\\caption{{\\protect\\footnotesize (Color online) The full width at half maximum $\\Delta \\alpha$ of the peak in entanglement around $\\alpha=\\pi$ at the optimal time versus the chain length.}}\n\\label{PeakWidth}\n\\end{figure}\n\n\\section{Series of Bell States as an Initial State}\n\\label{BS}\nThere may be other states which are easy to prepare as initial states in practice in various physical realizations. As long as their entanglement is not long range to start with, the quench induced dynamics still serves its purpose of generating long distance entanglement. We can thus look at an initial state which is a product of states where spins are entangled only with their nearest neighbors, namely a product of Bell states. We now consider the following product of Bell states as the initial state\n\\begin{equation}\n\\left|\\psi\\right\\rangle=\\mathop{\\otimes}_{k=1}^{N\/2} \\left|\\psi_{k,k+1}\\right\\rangle = \\mathop{\\otimes}_{k=1}^{N\/2} \\left(\\left|\\uparrow_{k} \\downarrow_{k+1}\\right\\rangle - \\left|\\downarrow_{k} \\uparrow_{k+1}\\right\\rangle \\right).\n\\end{equation}\nThe above state can be implemented using cold atoms in an optical superlattice formed by two independent lattices with different periods. Applying a spin-dependent energy offset results in pairs of singlets. \\citep{Yang,Rey}\\\\\nStarting with the above initial state, the system then evolves according to the $XX$ Hamiltonian (Eq.(\\ref{Hamiltonian}) with $\\Delta=0$). Using Eq.(\\ref{rho1N}) and\n\\begin{equation}\n\\left\\langle \\psi_{l,l+1} \\right|\\hat{\\sigma}_{l}^{\\pm}\\left|\\psi_{l,l+1}\\right\\rangle=0, \\left\\langle \\psi_{l,l+1} \\right|\\hat{\\sigma}_{l+1}^{\\pm}\\left|\\psi_{l,l+1}\\right\\rangle=0, \\left\\langle\\hat{\\sigma}_{1}^{\\pm}\\hat{\\sigma}_{N}^{\\pm} \\right\\rangle=0,\n\\end{equation}\nwe find that the only non-vanishing elements of the reduced density matrix are\n\\begin{equation}\n\\hat{\\rho}_{1,N}(t)=\\left(\\begin{array}{cccc}\na(t) & & & \\\\\n & b(t) & c(t) & \\\\\n & c'(t) & b'(t) & \\\\\n& & & a'(t)\n\\end{array}\\right) \\, .\n\\end{equation}\nThe matrix elements are\n\\begin{eqnarray}\na(t)&=&\\left\\langle \\hat{c}_{N}^{\\dagger}(t)\\hat{c}_{N}(t) \\right\\rangle \\left\\langle \\hat{c}_{1}^{\\dagger}(t)\\hat{c}_{1}(t) \\right\\rangle - \\left\\langle \\hat{c}_{N}^{\\dagger}(t)\\hat{c}_{1}(t) \\right\\rangle \\left\\langle \\hat{c}_{1}^{\\dagger}(t)\\hat{c}_{N}(t) \\right\\rangle\\\\\na'(t)&=&1 - \\left\\langle \\hat{c}_{1}^{\\dagger}(t)\\hat{c}_{1}(t) \\right\\rangle- \\left\\langle \\hat{c}_{N}^{\\dagger}(t)\\hat{c}_{N}(t) \\right\\rangle + a (t)\\\\\nb(t)&=&\\left\\langle \\hat{c}_{1}^{\\dagger}(t)\\hat{c}_{1}(t)\\right\\rangle - a(t)\\\\\nb'(t)&=&\\left\\langle \\hat{c}_{N}^{\\dagger}(t)\\hat{c}_{N}(t)\\right\\rangle - a(t)\\\\\nc(t)&=&\\sum_{l,m} f_{1,l}(t) f_{N,m}^{*} (t) \\left\\langle \\hat{c}_l (0)\\hat{c}_{m}^{\\dagger} (0) S_{1,N}\\right\\rangle\\\\\nc'(t)&=&\\sum_{l,m} f_{N,l}(t) f_{1,m}^{*} (t) \\left\\langle \\hat{c}_l (0)\\hat{c}_{m}^{\\dagger} (0) S_{1,N}\\right\\rangle\n\\end{eqnarray}\nFor the two point correlation function $\\left\\langle \\hat{c}_{i}^{\\dagger}(t)\\hat{c}_{j}(t)\\right\\rangle=\\sum_{l,m} f_{i,l}^{*}(t) f_{j,m}(t) \\left\\langle \\hat{c}_{l}^{\\dagger}(0)\\hat{c}_{m}(0)\\right\\rangle$ we have 3 cases, first if $|{l-m}|>1$, then\\begin{equation} \\left\\langle \\hat{c}_{l}^{\\dagger}(0) \\hat{c}_m (0) \\right\\rangle=0.\\end{equation}\nSecond, if $l=m$, then\n\\begin{equation}\n\\left\\langle \\hat{c}_{l}^{\\dagger}(0) \\hat{c}_l (0) \\right\\rangle=\\left\\langle \\hat{\\sigma}_{l}^{+}(0) \\hat{\\sigma}_{l}^{-}(0)\\right\\rangle=\\frac{1}{2}\n\\end{equation}\nFinally, if $|{l-m}|=1$, then\n\\begin{equation}\n\\left\\langle \\hat{c}_{l}^{\\dagger} (0) \\hat{c}_m (0) \\right\\rangle=\\left\\langle \\hat{\\sigma}_{l}^{+} (0)  \\hat{\\sigma}_{m}^{-} (0) \\right\\rangle=-\\frac{1}{2}\n\\end{equation}\nWhile for the term $\\left\\langle \\hat{c}_l (0)\\hat{c}_{m}^{\\dagger} (0) S_{1,N}\\right\\rangle$ we have 4 cases, first if $l=m$, then\n\\begin{equation}\n\\left\\langle \\hat{c}_{l}(0) \\hat{c}_{m}^{\\dagger} (0) \\hat{S}_{1,N} \\right\\rangle=(-1)^{N\/2-1} \\left\\langle \\hat{\\sigma}_{l}^{-} \\hat{\\sigma}_{l}^{+} \\hat{S}_{l+1}\\right\\rangle=\\frac{1}{2} (-1)^{N\/2}.\n\\end{equation}\nSecond, if $l=odd$ and $m=l+1$, then\n\\begin{equation}\n\\left\\langle \\hat{c}_{l}(0) \\hat{c}_{m}^{\\dagger} (0) \\hat{S}_{1,N} \\right\\rangle=\\left\\langle \\hat{\\sigma}_{l}^{-} \\hat{\\sigma}_{l+1}^{+} \\hat{S}_{1,l-1} \\hat{S}_{l+2,N} \\right\\rangle=\\frac{1}{2} (-1)^{N\/2}.\n\\end{equation}\nThird, if $l=even$ and $m=l-1$, then\n\\begin{equation}\n\\left\\langle \\hat{c}_{l}(0) \\hat{c}_{m}^{\\dagger} (0) \\hat{S}_{1,N} \\right\\rangle=\\left\\langle \\hat{\\sigma}_{l}^{-} \\hat{\\sigma}_{l-1}^{+} \\hat{S}_{1,l-2} \\hat{S}_{l+1,N} \\right\\rangle=\\frac{1}{2} (-1)^{N\/2}.\n\\end{equation}\nFinally, for all other cases\n\\begin{equation}\n\\left\\langle \\hat{c}_{l}(0) \\hat{c}_{m}^{\\dagger} (0) \\hat{S}_{1,N} \\right\\rangle=0.\n\\end{equation}\n\n\\begin{figure}[htbp]\n \\centering \n\\includegraphics[width=14cm]{Fig5.pdf}\n   \\caption{{\\protect\\footnotesize (Color online) Dynamics of the concurrence and the fully entangled fraction, of spins $1$ and $N$ of the chain for $N=24$ spins with a series of Bell pairs as an initial state. The scaled time $t$ is in units of $1\/J$.}}\n\\label{Singlets}\n\\end{figure}\n\n\\begin{figure}[htbp]\n \\centering \n\\includegraphics[width=14cm]{Fig6.pdf}\n   \\caption{{\\protect\\footnotesize (Color online) Dynamics of the concurrence and the fully entangled fraction, of spins $1$ and $N$ of the chain for $N=50$ spins with a series of Bell pairs as an initial state. The scaled time $t$ is in units of $1\/J$.}}\n\\label{Singlets50}\n\\end{figure}\n\nFigs.~\\ref{Singlets} and \\ref{Singlets50} show the dynamics of the concurrence and the fully entangled fraction of the distant end spins of the chain for $N=24$ spins, i.e. $12$ Bell pairs and $N=50$ spins, i.e., $25$ Bell pairs respectively. We note that the behavior is similar to the case of N\\'eel state. It shows that the initial state of a series of Bell pairs act very similarly to the N\\'eel state in terms of entanglement generation (maximally entangled fractions of $0.73$ and $0.65$ at times close to (slightly higher than) $t_{\\text{opt}}\\sim \\frac{N}{4J}$.\n\n\\begin{figure}[htbp]\n \\centering \n\\includegraphics[width=14cm]{Fig7.pdf}\n   \\caption{{\\protect\\footnotesize (Color online) (a) maximum fully entangled fraction of $\\rho_{1,N}$ versus the chain length, (b) the time at which the first peak of entanglement occurs versus the chain length.}}\n\\label{Scaling}\n\\end{figure}\n\n\\begin{figure}\n \\centering \n\\includegraphics[width=12cm]{Fig8.pdf}\n   \\caption{{\\protect\\footnotesize This figure shows the quantum walk of excitations (fermions) starting from an initial state (Eq.~\\ref{CantedState}). The $k=7$ fermion is shown to generate an equal distribution of probabilities at either end at time $t\\sim \\frac{N}{4J}$, whereas the $k=3$ fermion generates a state with unequal probabilities at the ends.}}\n \\label{expl}\n\\end{figure}\n\n\n\\section{Errors}\nIn any experimental implementation, some disorders will be present and might affect the resulting entanglement \\cite{Tsomokos}. To measure the robustness of our scheme we consider two types of disorder. First we study the effect of a small disorder in the preparation of the initial N\\'eel state. We consider a random single spin flip with a probability $N\\epsilon$, i.e. the initial density matrix is\n\\begin{equation}\n\\rho=\\left(1-N\\epsilon\\right)\\left|\\textrm{N\\'eel}\\right\\rangle\\left\\langle \\textrm{N\\'eel}\\right| + \\epsilon \\sum_{k=1}^{N} \\sigma_k^x\\left|\\textrm{N\\'eel}\\right\\rangle\\left\\langle \\textrm{N\\'eel}\\right| \\sigma_k^x\n\\end{equation}\nFigure~\\ref{CantedDisorder} shows the resulting fully entangled fraction for $N=24$ and $N=50$ with probabilities $N\\epsilon=0,0.05,0.1,0.15$ (top to bottom). The entanglement at $T_{max}$ is not seriously affected with such disorder with a spin flip probability as large as $0.05$ and $T_{max}$ is also unchanged. In \\cite{Wichterich} it was shown that this scheme for generating entanglement is also robust to randomness in the couplings throughout the chain for an initial N\\'eel state. We consider the effect of such disorder for an initial series of Bell pairs. The coupling between neighboring sites $J$ is taken to be $J_k=J(1+\\delta_k)$ with a normally distributed set $\\delta_k$ with zero mean value and standard deviation $\\delta$. Figure~\\ref{SingletsDisorder} shows the resulting entanglement for $N=10$ and $\\delta=0,0.1,0.2$ (top to bottom). The entanglement is not seriously suppressed for an average offset as large as $10\\%$ of the $J$ and $T_{max}$ is nearly unchanged.\n\n\\section{Experimental Implementations}\n\\label{impl} Spin models with the possibility of long time coherent dynamics are now being realized in some physical systems such as ultracold atoms in optical lattices and trapped ions. Here we outline the schematics of an implementation with ultracold atoms in optical lattices. Firstly a spin chain in an appropriate state has to be prepared. 1D tubes of light trapping atoms in a lattice are now quite routine \\cite{impurity,bound}. The initial state can be prepared by dimerizing the lattice to a series of double wells. Simply dimerizing it would produce a series of Bell states as in Ref.\\cite{Trotzky}, which would be the starting state for the scheme discussed in section \\ref{BS}. Further, by having a magnetic field gradient between the two wells of the lattice, one can also prepare a series of $|\\uparrow_k \\downarrow_{k+1} \\rangle$ or $|\\downarrow_k \\uparrow_{k+1} \\rangle$ in double wells, and thereby the spin texture of one of the N\\'eel states when many such wells are arranged in a series in a superlattice \\cite{Trotzky}. The local control which is now possible in combination of digital spatial light modulators and microwave pulses \\cite{impurity} should also enable one to prepare arbitrary spin textures if the results of section \\ref{canted} are to be verified. The sudden quenching (switching on) of the interactions is done by lowering the barriers between neighboring wells at time $t=0$ in a time-scale much faster (say, a $ms$) than $\\hbar\/J$, which has been achieved quite recently \\cite{impurity,bound}. In our setup, an open ended spin chain (in other words, a hard wall boundary) is required for the reflections. This should be achievable quite soon using the technique of digital light modulators which can eventually influence potentials at the scale of lattice site separations \\cite{impurity}. The readout of spins of the end sites to verify the entanglement should be achievable using quantum gas microscope technique \\cite{impurity}.\n\\section{Discussion and Summary}\nEntangling distant spins is desirable for connecting separated quantum registers through teleportation. We find that it can be achieved by exploiting the dynamics of a spin chain after a quench. We find that obtaining a high entanglement can depend sensitively on the initial magnetic order. For example, an initial N\\'eel state is an excellent resource, while more general canted order states hardly give any entanglement. Additionally, we found that a state composed of a product of singlets represent an excellent resource for generating entanglement when quenching the system by the $XX$ Hamiltonian. Our scheme can be implemented experimentally using, for example, ultracold atoms in optical lattices in which most of the requirements have been achieved lately, and an outline for the implementation is discussed in section \\ref{impl}. Of course, the entanglement, while long range, is {\\em not distance independent} in this scheme -- it falls with the length $N$ of the chain. This dependence, as well as the time needed to achieve the peak in entanglement, is plotted in Fig.\\ref{Scaling}. Note that for an initial product of Bell states, for shorter chains, the entanglement produced is slightly lower than that produced from the N\\'eel state -- though it catches up asymptotically as $N$ increases. Moreover, note from the slope of the line in Fig.\\ref{Scaling}(b) that the time needed is always linear in $N$ i.e., $\\sim \\frac{N}{4J}$. With the velocity of a spin flip in the $XX$ chain being $\\sim 2J$, this is the time needed for a spin flip to travel half the length of the chain. Note also from Fig.\\ref{Scaling}(a) that even for very large chains ($N >100$) the maximally entangled fraction remains above $0.5$ (i.e., still useful for distillation).\n\nBefore concluding we provide the reader with some intuitive understanding of our results. Firstly, let us first clarify the non-triviality of our results. Basically, it is known (and expected) that a non-zero entanglement will develop between complementary blocks of a spin chain after a quench, simply because of entangled quasiparticles crossing the boundary between the blocks \\cite{Calabrese,deChiara}. However, the spins at the two ends are ``non-complementary\" parts of the chain. The part of the chain between the two spins serves as an environment. Thus for a state of the ends to be significantly entangled, it is required that the intervening chain be in the {\\em same state} corresponding to distinct states of the end spins. To further clarify the situation, note that the Hamiltonian here conserves the number of spin flips. Thus states $|00\\rangle$ and $|11\\rangle$ of the end spins cannot have any coherence between them as they necessarily involve a different number of flips in the part of the chain between the spins. The only possibility of coherence is between the states $|01\\rangle$ and $|10\\rangle$ of the spin chain. However, though dynamics may produce the states $|01\\rangle$ and $|10\\rangle$ at the end spins, it is not guaranteed that the rest of the spin chain will be in the same state irrespective of whether the state $|01\\rangle$ or $|10\\rangle$ is generated between the end spins -- without that, there would be no entanglement between the end spins. It is highly nontrivial to ensure that the dynamics results in the central part of the chain being in the same state corresponding to $|01\\rangle$ and $|10\\rangle$ states. Because of reasons presented below, the Hamiltonian governing our dynamics and our initial state ensures this. It is additionally nontrivial that the amount of the entanglement between the end spins would be high. Why this is the case is explained in the following paragraphs.\n\n\n\nWe now proceed to explain the result that apart from states very close to the N\\'eel state, any other canted order state gives a vanishing entanglement and fully entangled fraction. This is because of the principal mechanism through which entanglement is generated from the N\\'eel state. For simplicity, we will present our explanation for odd $N$ chains. As shown in Fig.\\ref{expl}, each up spin is the location of a fermion. The evolution is due to a free fermion model, so each fermion starts evolving in the chain as if other fermions were absent except for the phase factor the wave-function acquires upon pairwise exchange of fermions. Each fermion evolves in a superposition of left and right moving fermions, doing a so called quantum walk on the chain. This moving in a superposition of left and right creates, for example, from a configuration $|010\\rangle$ in 3 successive sites a configuration of the form $|001\\rangle+|100\\rangle$, which, after factoring out the $|0\\rangle$ of the central site, is an entangled state $|01\\rangle+|10\\rangle$. This is how each fermion acts as a source of an entangled state in the chain \\citep{deChiara}. We will now justify why the simultaneous walk of all the fermions together then creates a highly entangled state between sites $1$ and $N$ at an optimal time. At time $t\\sim \\frac{N}{4J}$, where $2J$ is the velocity of fermions, the fermion at the exact centre of the chain will create, with some finite probability, a Bell state $|01\\rangle+|10\\rangle$ between the sites $1$ and $N$ because of the symmetry of the left and right walk. This is the $k=7$ fermion in Fig.\\ref{expl}, which shows the distribution of this fermion after a quantum walk for a time $t\\sim \\frac{N}{4J}$. Note, from Fig.\\ref{expl} that this fermion swaps its position with an equal number (in this case 3) of other fermions during its traversal to both the left and the right ends of the chain. Thus the relative phase between the left (i.e., $|10\\rangle$) and the right (i.e., $|01\\rangle$) components is $0$. The same holds for a fermion originating at any other site of the chain as both the left moving and the right moving components cross either an even number or an odd number of fermions (for example, in Fig.\\ref{expl} the $k=3$ fermion crosses 1 fermion on the left and 5 fermions on the right to reach the ends). Moreover, the initial placement of fermions on odd sites of the chain also ensures that the relative phase between the single fermion transition amplitudes ($f_{k,1}$ and $f_{k,N}$) from their original site to the left and the right ends (when no other fermion is present) is also $0$ as each fermion acquires a factor of $e^{i\\pi\/2}$ per hop \\footnote{This can be easily appreciated from the asymptotic form of the transition amplitudes $f_{kl} (t \\sim \\frac{N}{4J})\\sim i^{k-l}J_{k-l}(t \\sim \\frac{N}{4J})$.}. For example, the $k=3$ fermion shown in Fig.\\ref{expl} has 2 hops to reach the left end, giving a phase $e^{i\\pi}$, while it has 10 hops to the right end, which gives a phase $e^{i 5 \\pi}=e^{i \\pi}$. Thus the fermions originating on an arbitrary site of the chain, would, in general, contribute entangled states of the form $\\beta_1|01\\rangle+\\beta_N|10\\rangle$ to the end sites (this contribution, in general, could be mixed with $|00\\rangle$ and $|11\\rangle$), where $\\beta_1$ and $\\beta_N$ can both be taken to be real and positive (the global phase outside $\\beta_1|01\\rangle+\\beta_N|10\\rangle$ does not matter as each fermion starts from a distinct location and has a distinct evolution, so that the entangled state stemming from each fermion is incoherently added up to those from the others to generate the final state of the two end spins). In general $\\beta_1 \\neq \\beta_N$, as clear for the $k=3$ fermion in Fig.\\ref{expl} (only for the central fermion, for example for $k=7$ in Fig.\\ref{expl}, is $\\beta_1=\\beta_N$). Thus the coefficient of the off-diagonal term $|01\\rangle \\langle 10|$ contributed by each fermion is $\\beta^{*}_1 \\beta_N$, which is positive, and add up together to make a substantial off-diagonal term of the total density matrix. The entanglement is substantial because this off-diagonal term is substantial. Now imagine that there is a small probability of $\\epsilon$ for a fermion to be absent from a site were it was supposed to be present. This is a small deviation from the N\\'eel state. This causes a state of the form $\\beta_1|01\\rangle-\\beta_N|10\\rangle$ with $\\beta_1$ and $\\beta_N$ both real and positive, to be generated because if the left moving component of a fermion's state crosses an even number of other fermions then the right moving component crosses an odd number of other fermions and vice-versa. In Fig.\\ref{expl}, for example, if the fermion at the $k=5$ site was absent, then the central $(k=7)$ fermion would cross 2 fermions on its left, while it would still cross 3 fermions to its right. This will reduce the off-diagonal term by subtracting a proportion $\\epsilon \\beta_1^*\\beta_N$ from it. A very similar result holds when an extra fermion is present at a site were it was not supposed to be. Imagine, for example, the state of the $k=4$ site in Fig.\\ref{expl} was up instead of being as depicted. While this extra fermion itself will still create a state $\\beta_1|01\\rangle+\\beta_N|10\\rangle$, as can be verified by counting its number of hops and fermion crossings in reaching the ends, {\\em all} other fermions will now contribute a state\n$\\beta_1|01\\rangle-\\beta_N|10\\rangle$, by virtue of one extra fermion being now being crossed on either its leftward or its rightward walk to the ends. Thus overwhelmingly the presence of an extra fermion still results in contributions $-\\epsilon \\beta^{*}_1 \\beta_N$ to the original state. As there are $N$ sites in which a single spin flip error can occur, as soon as $\\epsilon \\sim \\frac{1}{N}$, the contributions of the form $-\\beta_1^{*}\\beta_N$ become of the order of unity and cancel contributions of the form $\\beta^{*}_1\\beta_N$ arising from the error-free N\\'eel state. Thus for very low errors of the magnitude $\\epsilon \\sim \\frac{1}{N}$ about the N\\'eel state, the entanglement completely vanishes. The canted ordered state, in fact, can be regarded as quite a systematic error whose probability $\\epsilon$ increases from the left to the right of the chain, and thereby $\\Delta \\alpha$ falls so rapidly with $N$. Thus the generated entanglement is quite rapidly lost as canting angles start deviating from the N\\'eel ordered state. The starting resource of Bell states is a case were we are starting with states $|01\\rangle-|10\\rangle$ on neighboring sites, which is as entangled as the state that is created after a one step (left and right in superposition) evolution of a single fermion from the N\\'eel state. The similarities of the resources near the starting point of the evolution is thereby responsible for both the product of Bell states and the N\\'eel state being excellent resources for establishing entanglement.\n\n\n\n\\begin{figure}\n \\centering \n\\includegraphics[width=12cm]{Fig9.pdf}\n   \\caption{{\\protect\\footnotesize (color online) The fully entangled fraction for an initial N\\'eel state with a random single spin flip with probability $N\\epsilon=0,0.05,0.1,0.15$ (top to bottom) for (a) $N=24$, (b) $N=50$.}}\n \\label{CantedDisorder}\n\\end{figure}\n\n\\begin{figure}\n \\centering \n\\includegraphics[width=12cm]{Fig10.pdf}\n   \\caption{{\\protect\\footnotesize (color online) The concurrence and fully entangled fraction for a series of Bell pairs as an initial state with random offsets in the couplings through the chain $\\delta=0,0.1,0.2$ (top to bottom) for $N=10$ (average taken over $100$ realizations).}}\n \\label{SingletsDisorder}\n\\end{figure}\n\n\\section{Acknowledgements} BA is supported by King Saud University. HW was supported by a UCL PhD plus fellowship when this work was started. The work of SB is supported by the ERC grant PACOMANEDIA. We wish to thank Abolfazl Bayat and Marcus Cramer for discussions that form a background to this work.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
{"text":"\\section{Definition of Ito\/Stratonovich integrals and conversion relations}\nWe report here the standard definitions of the Ito and Stratonovich stochastic integrals over a Wiener process $W_t$~\\cite{gardiner1985handbook}. For the Ito convention, we set for any sufficiently smooth function $g(t)$\n\\begin{subequations}\n \\label{itostrato}\n\\begin{align}\n \\int_0^t g(t) dW &\\equiv \\lim_{\\delta t \\to 0} \\sum_i g(t_{i-1}) (W_{t_i} - W_{t_{i-1}})\\;, \\quad \\mbox{Ito} \\\\\n \\int_0^t g(t) \\circ dW &\\equiv \\lim_{\\delta t \\to 0} \\sum_i g\\Bigl(\\frac{t_{i-1} + t_{i}}{2}\\Bigr) (W_{t_i} - W_{t_{i-1}})  \\;, \\quad \\mbox{Stratonovich}\n\\end{align}\n\\end{subequations}\nwhere $W_{t}$ denotes a Wiener process, with\n\\begin{equation}\n \\langle W_t \\rangle = 0 \\;, \\langle (W_{t} - W_{t'})^2 \\rangle = |t-t'|\n\\end{equation}\nOne should notice the difference between the two cases: in the second one, the function $g$ is computed at the middle point between the two different times $t$ and $t+ dt$. The main consequence is that it is not statistically independent from the increment $dW(t) \\sim W_{t + dt} - W_{t}$. Therefore, the Ito convention is preferred if one is interested in taking the noise average\n\\begin{equation}\n \\langle g(t) dW_t \\rangle = \\langle g(t) \\rangle \\langle dW_t \\rangle = 0\\, .\n\\end{equation}\n\nThe two conventions give rise to different equation of motions. However it is possible to convert one into the other. Consider two prototypical stochastic equations of the form in the Ito-Stratonovich conventions with a single variable $X$\n\\begin{subequations}\n\\label{sde}\n\\begin{align}\n& dX_t = f(X_t) dt + g(X_t) dW_t \\label{ito}\\\\\n& dX_t = \\tilde f(X_t) dt + \\tilde g(X_t) \\circ dW_t \\label{strato}\\, ,\n\\end{align}\n\\end{subequations}\nwhich in the integrated form they become respectively\n\\begin{align}\n& X(t) = X(0) + \\int_0^t \\, ds \\; f(X(s))  + \\int_0^t g(X(s)) dW\\\\\n& X(t) = X(0) + \\int_0^t \\, ds \\; \\tilde f(X(s))  + \\int_0^t \\tilde g(X(s)) \\circ dW\\, .\n\\end{align}\nOne can show that Eqs.~\\eqref{sde} are actually describing the same stochastic process if \n\\begin{equation}\n\\label{eq:itotostrato}\n \\tilde g(X) = g(X) \\;, \\qquad \\tilde f(X) = f(X) - \\frac{1}{2} g(X) g'(X)\\, .\n\\end{equation}\n\n\\subsubsection{Multi-dimensional case}\nThis formula admits a simple generalizaton to the vector case. We replace Eq. \\eqref{sde} with\n\\begin{subequations}\n\\label{sdevec}\n\\begin{align}\n& dX_\\mu(t) = f_\\mu(X_t) dt + g_{\\mu,a}(X_t) dW_a(t) \\label{itovec}\\\\\n& dX_\\mu(t) = \\tilde f_\\mu(X_t) dt + \\tilde g_{\\mu,a}(X_t) \\circ dW_a(t) \\label{stratovec}\n\\end{align}\n\\end{subequations}\nwhere $X_\\mu$ is now a $N$-component vector indexed by $\\mu=1,\\ldots N$ and $dW_a$ are $M$ independent Wiener processes\n\\begin{equation}\n \\langle dW_a(t) dW_{a'}(t) \\rangle = \\delta_{aa'} dt\\, .\n\\end{equation}\n\nThe sum over repeated indexes is assumed. In this case, one has the relations\n\\begin{equation}\n\\label{itostratovec}\n \\tilde g_{\\mu, a}(X) = g_{\\mu,a}(X) \\;, \\qquad \\tilde f_\\mu(X) = f_\\mu(X) - \\frac{1}{2} \\partial_\\nu g_{\\mu,a}(X) g_{\\nu,a}(X) \\, .\n\\end{equation}\n\n\n\\section{Derivation of the Lindblad equation}\nIt is useful to rewrite the noise by introducing a set of indepedent processes. Given the noise correlation\n\\begin{equation}\n\\langle \\eta_j(t) \\eta_{j'}(t') \\rangle = \\gamma F(j-j') \\delta(t-t')\n\\end{equation}\nwe define a linear combinaton of the noise which diagonalises the correlation, i.e. \n\\begin{equation}\n \\eta_j(t) dt = \\sqrt{\\gamma}\\sum_{j'} f(j-j') dW_j(t)  \\;, \\quad \\sum_k f(j-k) f(j'-k) = F(j-j')\n\\end{equation}\nwhich implies\n\\begin{equation}\n\\langle dW_j(t) dW_{j'}(t) \\rangle = \\delta_{j,j'} dt  \\, .\n\\end{equation}\nThe Schr\\\"odinger equation in Eq. \\eqref{stochschr} can then be written in the form\nEq. \\eqref{stratovec} as\n\\begin{equation}\n\\label{stochschrstrato}\nd\\ket{\\psi} = -\\imath \\hat{H}_{0}  \\ket{\\psi} dt -\\imath \\sqrt{\\gamma}\\sum_j  \\tilde q_j  \\ket{\\psi} \\circ dW_j(t) \\;, \\quad \\tilde q_j = \\left[\\sum_{j'} f(j-j') \\hatq_{j'}\\right]\\, ,\n\\end{equation}\nwhere $\\tilde q_j$ is a smoothened version of the local density $\\hat q_j$ via the convolution Kernel $f(j)$. We can now apply Eq.~\\eqref{itostratovec} to obtain the Ito form of the Schr\\\"odinger equation\n\\begin{equation}\n    d\\ket{\\psi} =\n    -\\imath H_0 \\ket{\\Psi} dt -\\frac  {\\gamma}  2\\sum_j \\tilde q_j^2 \\ket{\\Psi} dt - \\imath \\sqrt{\\gamma}\\sum_j \\tildeq_j   \\ket{\\Psi} dW_j(t)\\, .\n\\end{equation}\nFrom this, applying the Ito's lemma, we can get the average evolution of the density matrix\n\\begin{equation}\n\\frac{d\\bm{\\varrho}}{dt} = -\\imath [\\hat H_0, \\bm{\\varrho}] + \\gamma \\sum_j \\left(\\tilde q_j \\bm{\\varrho} \\tilde q_j - \\frac12 \\{ \\tilde q_j^2, \\bm{\\varrho} \\}\\right)\\, ,\n\\end{equation}\nwhich coincides with Eq.~\\eqref{eq:lindblad} in the main text.\n\n\n\\section{Derivation of the hydrodynamic equation}\n\nHere we provide a detailed derivation of the hydrodynamic equation \\eqref{eq:finaldiffusion2}. As we discussed in the main text, in the limit of weak noise $\\gamma\\ll 1$ and large correlation length $\\ell \\gg 1$, for any charge $\\bm{Q}^{(\\alpha)}$, Eq. \\eqref{eq:chargevar} can be written as\n\\begin{equation}\n\\lim_{L \\to \\infty} \\frac{\\operatorname{Tr}[\\bm{Q}^{(\\alpha)} \\dot{\\bm{\\bm{\\varrho}}}]}{L} = \n\\int \\lambda q^{(\\alpha)}\\partial_t \\rho(\\lambda)=-\\frac{\\gamma \\kappa_2}{\\ell}  \\int d\\lambda \\, (\\partial_\\lambda q^{\\rm dr}\\,^{(\\alpha)}(\\lambda)) [V^{\\bm{O}}(p(\\lambda)]^2\n \\partial_p n(p(\\lambda))\\, .\n\\end{equation}\n\nWe now integrate by parts using the definition of dressing\n\\begin{multline}\n-\\int d\\lambda \\,\\partial_\\lambda q^{\\text{dr}(\\alpha)}(\\lambda)[V^{\\bm{O}}(p(\\lambda)]^2 \\partial_p n(p(\\lambda))=-\\int d\\lambda\\int d \\lambda' \\partial_{\\lambda'}q^{(\\alpha)}(\\lambda')(1+nT)^{-1}_{\\lambda',\\lambda}[V^{\\bm{O}}(p(\\lambda)]^2 \\partial_p n(p(\\lambda))=\\\\ \\int d\\lambda\\int d \\lambda' q^{(\\alpha)}(\\lambda')\\partial_{\\lambda'}\\Bigg[(1+nT)^{-1}_{\\lambda',\\lambda}[V^{\\bm{O}}(p(\\lambda)]^2 \\partial_p n(p(\\lambda))\\Bigg]\\, ,\n\\end{multline}\nthen we use the completeness of the charges $\\bm{Q}^{(\\alpha)}$ to extract from the infinitely-many integral equations a differential equation for the root density.\nIn the rapidity space we obtain\n\\begin{equation}\\label{Seq_ghd_rho}\n\\partial_t \\rho(\\lambda)=\\frac{\\gamma \\kappa_2}{\\ell}\\partial_{\\lambda} \\Bigg[(1+nT)^{-1}[V^{\\bm{O}}]^2 \\partial_p n\\Bigg]\\, .\n\\end{equation}\n\nThis equation can be greatly simplified if written in terms of the filling. Indeed, using $\\rho(\\lambda)=n(p(\\lambda))\\rho_t(\\lambda)$, one can readily show\n\\begin{equation}\n\\partial_t \\rho(\\lambda)=(1+n T)^{-1}(\\rho_t\\partial_t n)\\, ,\n\\end{equation}\nwhich leads to the equation\n\\begin{equation}\\label{eq_S21}\n\\partial_t n(p(\\lambda))=\\frac{\\kappa_2 \\gamma}{\\ell\\rho_t(\\lambda)}\\Bigg[\\partial_\\lambda( [V^{\\bm O}]^2\\partial_p n)-\\partial_\\lambda n(p(\\lambda))T^{\\text{dr}} ([V^{\\bm O}]^2\\partial_p n))\\Bigg]\\, .\n\\end{equation}\nAbove, we used the definition of the scattering kernel $T(1 + n T)^{-1} = (1+ T n)^{-1} T = T^{\\rm dr}$. We finally pass to the momentum space: while performing this operation, one should not forget the time dependence of the mapping between the rapidity and momentum\n\\begin{equation}\\label{eq_S22}\n\\partial_t n(p(\\lambda))=\\partial_t n(p)+\\partial_t p(\\lambda) \\partial_p n(p)\\, .\n\\end{equation}\nWe now use that the total root density is the dressed derivative of the momentum\n\\begin{equation}\n\\rho_t(\\lambda)= \\frac{\\partial_\\lambda p_\\text{bare}(\\lambda)}{2\\pi}-\\int d \\lambda' T(\\lambda,\\lambda')\\rho(\\lambda')\\, ,\n\\end{equation}\nwhere $p_\\text{bare}(\\lambda)$ is the bare momentum, which is state independent. Taking the time derivative of the above and using \\eqref{Seq_ghd_rho} we get\n\\begin{equation}\n\\partial_t\\rho_t(\\lambda)=\\frac{\\gamma \\kappa_2}{\\ell} \\int d \\lambda' T(\\lambda,\\lambda')\\partial_{\\lambda'} \\Bigg[(1+nT)^{-1}[V^{\\bm{O}}]^2 \\partial_p n\\Bigg]=\\frac{\\gamma \\kappa_2}{\\ell}\\partial_\\lambda T^{\\text{dr}}[V^{\\bm{O}}]^2 \\partial_p n\\, .\n\\end{equation}\nIn the last passage, we integrated by parts (assuming that the boundary terms can be neglected) and used the symmetry of the kernel. \nUsing $dp=2\\pi \\rho_t d \\lambda$ and integrating the above equation, we find\n\\begin{equation}\\label{eq_S25}\n\\partial_t p(\\lambda)= \\int_{\\lambda_0}^\\lambda d\\lambda' 2\\pi\\partial_t \\rho_t(\\lambda') = \\frac{\\gamma \\kappa_2}{\\ell}2\\pi T^{\\text{dr}}[V^{\\bm{O}}]^2 \\partial_p n\\Big|_{\\lambda}-\\frac{\\gamma \\kappa_2}{\\ell}2\\pi T^{\\text{dr}}[V^{\\bm{O}}]^2 \\partial_p n\\Big|_{\\lambda_0}\\, .\n\\end{equation}\nAbove, we explicitly wrote both the integration boundaries. In several instances, the rapidity $\\lambda_0$ can be chosen in such a way that $2\\pi T^{\\text{dr}}[V^{\\bm{O}}]^2 \\partial_p n\\Big|_{\\lambda_0}=0$. For example, this is true for parity invariant states $\\lambda\\to -\\lambda$ if we set $\\lambda_0$. Assuming this simplification and combining Eqs. \\eqref{eq_S22} and \\eqref{eq_S25} in Eq. \\eqref{eq_S21}, we obtain Eq. \\eqref{eq:finaldiffusion2} of the main text.\nFor the numerical solution of the hydrodynamic equation, we found that the most stable and efficient approach was to solve directly Eq. \\eqref{eq_S21} directly in the rapidity space. As it is typical in diffusion-like equations (involving second derivatives), the Crank-Nicholson algorithm \\cite{thomas2013numerical} provided the sufficient stability to evolve the equation at large times.\n\n\\section{Noise average over the force field }\n\nWithin this section, we work in the rapidity space and for the sake of simplicity we define the filling $n(\\lambda) \\equiv n(p(\\lambda))$  rather than the one in terms of the dressed momenta $p$. \nWe consider the equivalent formulation of the hydrodynamic equation \\eqref{eq_S21}  (in the case where the driving is coupled to a conserved charge $\\bm O=\\bm q$)\n\n\\begin{equation}\\label{eq:motionolambda}\n\\partial_t n_{t}  = \\frac{\\kappa_2 \\gamma}{\\ell}\n  \\left[ \\frac{1}{{{p}'} }  { \\partial_\\lambda \\left(\\frac{n'   ({{q}}^{\\rm dr} )^2}{{{p}'}} \\right)-   n' T^{\\rm dr} \\frac{(q^{\\rm dr})^2}{p' }  n' }{}  \\right]_t,\n\\end{equation}\nwhere $n' = \\partial_\\lambda n_t(\\lambda)$ and where the right hand side is evaluated at time $t$. Notice that we used the general notation\n\\begin{equation}\ng_1 T^{\\rm dr} g_2= \\int d\\lambda' g_1(\\lambda) T^{\\rm dr}(\\lambda,\\lambda') g_2(\\lambda').\n\\end{equation}\n  \nWe now wish to show that the dynamics given by a GHD with a random force, namely :\n\\begin{equation}\nd n_{x,t}(\\lambda) +  v^{\\rm eff}_{x,t}[\\varepsilon+U q] \\partial_x n_{x,t}(\\lambda) dt = \\sqrt{\\gamma}  \\frac{q^{\\rm dr}_{x,t}(\\lambda)}{p'_{x,t}(\\lambda)} \\partial_\\lambda n_{x,t}(\\lambda)  \\partial_x \\sum_{x'} F({x-x'})dW_{x'}(t),\n\\end{equation}\nwhere $v^{\\rm eff}[\\varepsilon+U q] =   v^{\\rm eff}_{x,t}[\\varepsilon] + v^{\\rm eff}_{x,t}[q]$ is given as the sum of two effective velocities, one given by the dispersion relation of the quasiparticles $\\varepsilon(p)$ and one given by the correction to the energy given by the external potential\n\\begin{equation}\nv^{\\rm eff}_{x,t}[\\varepsilon](\\lambda) =  \\frac{ \\varepsilon' (\\lambda)}{p'(\\lambda)} \\quad ,\\quad v^{\\rm eff}_{x,t}[q]  (\\lambda)  =  \\frac{\\sum_{x'} F({x-x'})\\eta(x',t) (q')^{\\rm dr}(\\lambda) }{p'(\\lambda)}  .\n\\end{equation}\n\nIn order to average over the noise, we first need to apply Eq.~\\eqref{itostratovec} to convert the noise in the Ito form, where the transformations are now defined as \n\\begin{align}\n& f_{x,\\lambda}(n) = -  v^{\\rm eff}_{x,t}[\\varepsilon]^{\\rm eff}(\\lambda) \\partial_x n_{x,t}(\\lambda) \\\\\n& g_{x,\\lambda, x'}(n) = - \\sqrt{\\gamma } v^{\\rm eff}_{x,t}[q]^{\\rm eff}(\\lambda)  F({x-x'}) \\partial_x n_{x,t}(\\lambda) \n+\\sqrt{\\gamma}  \\frac{q^{\\rm dr}_{x,t}(\\lambda)}{p'_{x,t} (\\lambda)} \\partial_\\lambda n_{x,t}(\\lambda)  \\partial_x F({x-x'})\n\\end{align}\nTherefore, relation \\eqref{itostratovec} reduces to computing the following, neglecting terms proportional to $\\partial_x n$, which will average to zero, \n\\begin{align}\n& \\int d\\lambda' dx' dx'' \\frac{\\delta g_{x,\\lambda, x'}}{\\delta n_{x'',t}( \\lambda')} g_{x'',\\lambda', x'}= \\int dx'\\Big(  - \\sqrt{\\gamma} {v^{\\rm eff}_{x,t}[q]^{\\rm eff}}(\\lambda) F({x-x'}) \\partial_x g_{x, \\lambda, x'}  \\nonumber \\\\& +   \\sqrt{\\gamma}  n'_{x,t}(\\lambda) \\int d\\lambda' \\frac{\\delta [q^{\\rm dr}_{x,t}(\\lambda)\/p'_{x,t}(\\lambda)]}{\\delta n_{x,t}(\\lambda')}  F'({x-x'}) g_{x,\\lambda',x'} -  \\sqrt{\\gamma} \\frac{q^{\\rm dr}_{x,t}(\\lambda)}{p'_{x,t}(\\lambda)} F'({x-x'}) \\partial_\\lambda g_{x,\\lambda,x'} \\Big),\n\\end{align}\nwhere we denotes with $F'$ and $n'$ the derivative respect too $x$ or $\\lambda$, respectively and we used \n\\begin{equation}\n\\frac{ \\delta n_{x,t}(\\lambda)}{ \\delta n_{x',t}(\\lambda')} = \\delta(\\lambda-\\lambda') \\delta(x-x').\n\\end{equation} \nThe functional derivative can be taken using the definition of dressing $q^{\\rm dr} = (1+  T n)^{-1} q$ and $p' = (1 + T n)^{-1} p'_{\\rm bare}$. We therefore have the following relation\n\\begin{equation}\n  \\frac{\\delta [q^{\\rm dr}\/p'^{\\rm dr}](\\lambda)}{\\delta n(\\lambda')}  =   \\frac{1}{p' (\\lambda)} T^{\\rm dr}(\\lambda,\\lambda')q^{\\rm dr}(\\lambda') + \\frac{q^{\\rm dr}(\\lambda)}{(p'(\\lambda))^2}T^{\\rm dr}(\\lambda,\\lambda') p'(\\lambda') .\n\\end{equation}\nPutting all terms together and finally averaging over the noise using \n\\begin{equation}\n \\langle dW_x(t) dW_{x'}(t) \\rangle = \\delta_{xx'} dt,\n\\end{equation}\nand the property\n\\begin{equation}\n\\int dx F(x) F''(x) = - \\int (F'(x))^2 = - 2 \\kappa_2 \\gamma \/\\ell,\n\\end{equation}\nwe can write the evolution of the averaged occupation function, as in Eq. \\eqref{strato}, as  \n\\begin{align}\\label{eq:interme}\n&\\partial_t n_t = \\frac{  \\kappa_2 \\gamma}{\\ell}  \\Big[ \\frac{(q')^{\\rm dr}  q^{\\rm dr}}{(p')^2}  n' + \\frac{q^{\\rm dr}}{p'} \\partial_\\lambda \\left( \\frac{q^{\\rm dr}}{p'}  n' \\right) -   n' T^{\\rm dr}   \\frac{  {(q^{\\rm dr})^2 }{ }  }{p'}  n'   -  \\frac{n'  \\ q^{\\rm dr}  }{( p')^2} { T^{\\rm dr} {q^{\\rm dr} }  n' }  . \\Big]_t\n\\end{align}\n In order to compare with Eq. \\eqref{eq:motionolambda} we first need then to express $(q')^{\\rm dr} $ in terms of $(q^{\\rm dr})'$. This can be done using \n\\begin{equation}\n(1 + T n ) q^{\\rm dr} =   q.\n\\end{equation}\nSo that by deriving left hand side and right hand side we obtain\n\\begin{equation}\n(q^{\\rm dr})'  + T^{\\rm dr} n' q^{\\rm dr} = (q')^{\\rm dr} ,\n\\end{equation}\nwhere we used that the scattering kernel is only a a function of the difference of rapidities $T(\\lambda,\\lambda') = T(\\lambda - \\lambda')$.\nTherefore the following relation is valid \n\\begin{equation}\n n' \\frac{(q'^{\\rm dr} ) q^{\\rm dr}}{(p')^2}   = n' \\frac{(q^{\\rm dr} )' q^{\\rm dr}}{(p')^2}  + \\frac{n'  q^{\\rm dr} }{(p')^2 }  {T^{\\rm dr} n' q^{\\rm dr} }{} .\n\\end{equation}\nThe second term in the above equation simplifies the last term in Eq. \\eqref{eq:interme} and we  therefore obtain our final equation \\eqref{eq:motionolambda}.\n\n\n\\section{Numerical methods for the classical interacting Bose gas} \n\nHere we shortly outline the method used for the numerical simulation of the classical interacting Bose gas.\nThe continuum system is discretized on a finite lattice of $N$ sites and lattice spacing $a$, periodic boundary conditions are assumed.\nThen, the algorithm consists in three steps.\n\\begin{enumerate}\n\\item \\emph{Sampling the thermal distribution with a Metropolis-Hasting method \\cite{10.1093\/biomet\/57.1.97}}\n\nWe consider $N$ complex variables $\\{\\psi_i\\}_{i=1}^N$ which are the discretized field. Then, we aim to sample thermal distributions, i.e. the probability of a certain field configuration $p[\\{\\psi_i\\}_{i=1}^N]$ obeys\n\\begin{equation}\\label{eq_pr_met}\np[\\{\\psi_i\\}_{i=1}^M]=\\frac{1}{\\mathcal{Z}}e^{- \\beta E[\\{\\psi_j\\}_{j=1}^N]}\\, , \\hspace{2pc} E[\\{\\psi_j\\}_{j=1}^N]= a\\sum_{j=1}^N \\left\\{\\frac{|\\psi_{j+1}-\\psi_j|^2}{a^2}+c|\\psi_j|^4-\\mu|\\psi_j|^2\\right\\}\n\\end{equation}\nAbove, periodic boundary conditions are assumed, $\\beta$ is the inverse temperature and $\\mu$ the chemical potential. The partition function $\\mathcal{Z}$ is needed for a correct normalization, but its value is not important.\nThen, the field configurations are updated according with an ergodic dynamics $\\{\\psi_i\\}_{i=1}^N\\to \\{\\psi_i'\\}_{i=1}^N$, whose fundamental step is divided into three parts \n\\begin{enumerate}\n\\item Choose at random a lattice site $j$.\n\\item Change the field in position $j$ as $\\psi_j\\to \\psi_j +\\delta \\psi_j$, with $\\delta \\psi_j$ a random gaussian variable of zero mean and variance $\\Omega=\\langle |\\delta\\psi_j|^2\\rangle$. The variance $\\Omega$ must be chosen in such a way that the acceptance rate (see below) is roughly $0.5$.\n\\item Compute the change in the Metropolis energy $\\delta E= E[\\{\\psi_j'\\}_{j=1}^N]- E[\\{\\psi_j\\}_{j=1}^N]$.\nIf $\\delta E<0$ the new configuration is accepted, otherwise it is accepted with probability $e^{-\\beta\\delta E}$.\n\\end{enumerate}\nThe Metropolis evolution is an ergodic process with respect to the probability \\eqref{eq_pr_met}, which can be sampled picking field configurations along the Metropolis evolution.\n\\item \\emph{Time evolve a sampled configuration with the microscopic equation of motion for a given noise's realization}\n\nThe discrete equation of motion (in the Ito convention) to be solved is\n\\begin{equation}\ni\\partial\\psi_j(t)=-a^{-2}(\\psi_{j+1}(t)+\\psi_{j-1}(t)-2\\psi_j(t))-c|\\psi_j(t)|^2\\psi_j(t)- \\eta_j(t)\\psi_j(t)\n\\end{equation}\nwith $\\eta_j(t)$ the driving. We now discretize the time on a finite grid with spacing $d t$ and define the fields $\\psi_j(s d t)\\to \\tilde{\\psi}_j(s)$ and $\\eta_j(sd t)\\to\\tilde{\\eta}_j(s)$. \nThe discrete noise is a random real gaussian variable with zero mean and correlation\n\\begin{equation}\n\\langle \\tilde{\\eta}_j(s) \\tilde{\\eta}_{j'}(s') \\rangle=\\frac{\\gamma}{a d t}\\delta_{j,j'}\\delta_{s,s'}F(a(j-j'))\\, .\n\\end{equation}\nThis scaling guarantees the correct continuous limit.\nIn order to obtain a stable evolution, we trotterize the dynamic evolving separately first with  the interaction and noise and then with the kinetic term. \nThe first part of the evolution states\n\\begin{equation}\n\\tilde{\\psi}_j(s+1\/2)=e^{-i dt (c|\\tilde{\\psi}_j(s)|^2+\\tilde{\\eta}_j(s))}\\tilde{\\psi}_j(s)\\, .\n\\end{equation}\nThe kinetic part is instead solved in the Fourier space $\\varphi_j(s)=\\sum_{j'}e^{i j j' 2\\pi\/N} \\tilde{\\psi}_{j'}(s)$, where it acts diagonally\n\\begin{equation}\n\\varphi_j(s+1)=\\exp\\left\\{-i 2a^{-2}\\left[1-\\cos\\left(2\\pi jN^{-1}\\right)\\right]\\right\\}\\varphi_j(s+1\/2)\\, .\n\\end{equation}\n\nThe above steps are then repeated to evolve the field up to the desired time.\n\\item \\emph{Average over the initial conditions and the noise realizations.}\n \nWe took advantage of the translation symmetry and perform spacial averaging as well.\n\\end{enumerate}\n\n\nIn our simulations we used $a=0.06$, $N=2^{10}$ and $dt=0.0125$: this choice guarantees us a good convergence within the desired precision.\nThe error bars are estimated running four independent and identical samplings and considering the variance. We collected roughly $3000$ samples in total.\n\n\n\\section{The semiclassical limit of the 1d Bose gas}\n\nThe thermodynamics of the quantum 1d interacting Bose gas in nowadays a textbook topic \\cite{takahashi2005thermodynamics}.\nWithin the repulsive phase $c>0$, the model is described by a single species of excitation (while in the attractive regime $c<0$ there exist infinitely many bound states) and the quantum scattering shift $T_\\text{q}(\\lambda,\\lambda')$ is\n\\begin{equation}\nT_\\text{q}(\\lambda,\\lambda')=-\\frac{1}{2\\pi}\\frac{2c}{(\\lambda-\\lambda')^2+c^2}\\, ,\n\\end{equation}\nfrom which the hydodynamics description follows. The energy, momentum and number of particles eigenvalues are $e(\\lambda)=\\lambda^2$, $p(\\lambda)=\\lambda$ and $N(\\lambda)=1$ respectively.\nOn the contrary, the classical model at finite energy density is far less studied, therefore we provide a short recap building on the findings of Ref. \\cite{vecchio2020exact}. The description of the classical 1d interacting Bose gas can be accessed through a proper \\emph{semiclassical limit} of the quantum model: the latter is achieved restoring $\\hbar$ in the quantum Hamiltonian \\cite{vecchio2020exact}\n\\begin{equation}\n\\hat{\\bm{H}}_0=\\int d x\\,\\left[\\hbar^2\\partial_x\\psi^\\dagger\\partial_x\\psi+c \\hbar^4\\,\\psi^\\dagger\\psi^\\dagger\\psi\\psi\\right]\\, ,\n\\end{equation}\nthen taking the limit $\\hbar\\to 0$: classical physics emerges from the quantum one in the limit of small interactions and high occupation numbers, as it is expected.\nSuch a limit results in the classical scattering shift\n\\begin{equation}\nT_\\text{cl}(\\lambda,\\lambda')=\\lim_{\\xi\\to 0^+}-\\frac{1}{2\\pi}\\frac{2c}{(\\lambda-\\lambda')^2+\\xi}\\, .\n\\end{equation}\nThe kernel is formally singular and in integral expressions the limit $\\xi\\to 0^+$ must be taken only after the integration. For any fast decay test function $\\tau(\\lambda)$ this amounts to the regularization\n\\begin{equation}\n\\int d \\lambda' \\, T_{\\text{cl}}(\\lambda,\\lambda')\\tau(\\lambda')=\\fint \\frac{d\\lambda'}{2\\pi} \\frac{2c}{\\lambda-\\lambda'}\\partial_{\\lambda'}\\tau(\\lambda')\\, ,\n\\end{equation}\nwhere on the r.h.s. the singular integral is regularized with the principal value prescription. Albeit there is not a formulation of the form factors within the classical context, the classical hydrodynamics can be obtained through the semiclassical limit of the quantum one, using the correspondence $\\lim_{\\hbar\\to 0} \\hbar^2 n_\\text{q}(\\lambda) \\tau^{\\text{dr}(q)}(\\lambda)=n_\\text{cl}(\\lambda) \\tau^{\\text{dr(cl)}}$ for an arbitrary test function $\\tau(\\lambda)$, where with $\\tau^{\\text{dr}(q)}$ and $\\tau^{\\text{dr}(cl)}$ we label the dressing performed with the quantum and classical kernel respectively.\nThe result of this limit is exactly Eq. \\eqref{eq:finaldiffusion2}, where the dressing is performed with the classical kernel. The energy, momentum and particles eigenvalues remain $e(\\lambda)=\\lambda^2$, $p(\\lambda)=\\lambda$ and $N(\\lambda)=1$ respectively.\nIn the main text, we chose as the initial state thermal ensembles, whose filling is determined by the following non-linear integral equation\n\\begin{equation}\n\\varepsilon(\\lambda)= \\beta\\big[e(\\lambda)-\\mu\\big]-\\int \\frac{d \\lambda'}{2\\pi}\\frac{2c}{\\lambda-\\lambda'}\\partial_{\\lambda'}\\log\\left( \\varepsilon(\\lambda')\\right)\n\\end{equation}\nwith $n_\\text{cl}=1\/\\varepsilon(\\lambda)$. Notice that with respect to the quantum case, the parametrization of the filling in terms of the pseudoenergy $\\varepsilon(\\lambda)$ is different: within the quantum case, the excitations follow a Fermi-Dirac distribution, while in the classical case one finds a Rayleigh-Jeans law. This is ultimately responsible for the UV divergence of the energy expectation value on thermal states, analogously to the famous UV-catastrophe in the black-body radiation.\nAs observables, we focused on the momenta of the density $|\\psi(x)|^{2n}$ and on the FCS of the density operator: these can be determined on arbitrary GGEs (hence at any time of the GHD solution) solving proper integral equations. The general expressions can be found in Ref. \\cite{vecchio2020exact}.\n\n\n\\section{The bosonization approach}\n\nThe Luttinger field theory \\cite{haldane1981demonstration,haldane1981luttinger,haldane1981effective,\nCazalilla_2004} is ubiquitous in describing the ground state and low energy excitations of 1d systems with a $U(1)$ conserved quantity, regardless the integrability of the model. This charge could be, for instance, the number of particles in the interacting Bose gas or the $z-$magnetization in the XXZ spin chain.\nTherefore, we can access the short-time behavior within bosonization, which must be in agreement with the short-time expansion of our hydrodynamic evolution.\nFluctuations of the conserved charge $\\delta \\bm{q}(x)$ are described in terms of a phase field $\\hat{\\phi}(x)$ with the correspondence $\\delta \\bm{q}(x)=-\\frac{1}{\\pi} \\partial_x\\hat{\\phi}$: we focus only on a dephasing associated with the $U(1)$ charge, therefore the dynamics is governed by the following stochastic Hamiltonian\n\\begin{equation}\n\\bm{H}_0=\\frac{1}{2\\pi}\\int d x\\,  v\\Big[K \\pi^2 \\hat{\\Pi}^2+\\frac{1}{K}(\\partial_x\\hat{\\phi})^2\\Big]-\\int d x\\,  \\eta_x(x) \\frac{\\partial_x\\hat{\\phi}}{\\pi}+\\text{const.}\\, .\n\\end{equation}\nAbove, $\\hat{\\Pi}$ is the momentum conjugated to the phase field $[\\hat{\\phi}(x),\\hat{\\Pi}(y)]=i\\delta(x-y)$ and the noise $\\eta_x(t)$ is, as usual, gaussian and $\\delta-$correlated $\\langle \\eta_x(t)\\eta_{x'}(t')\\rangle=\\gamma\\delta(t-t')F(x-x')$. The parameters $v$ and $K$ are the sound velocity and the Luttinger parameter respectively: within generic models these can be numerically or experimentally extracted from the correlation functions, but in the case of integrable models they can be analytically determined \\cite{Cazalilla_2004}.\nIn the absence of noise, the Luttinger Hamiltonian is diagonalized in terms of bosonic modes $[\\hat{a}(k),\\hat{a}^\\dagger(q)]=2\\pi \\delta(k-q)$\n\\begin{equation}\n\\hat{\\phi}(t,x)=\\int \\frac{d k}{2\\pi}e^{ik x}\\sqrt{\\frac{\\pi K}{2 |k|}} \\big[e^{-i v|k| t}\\hat{a}(k)+e^{i v|k| t}\\hat{a}^\\dagger(-k)\\big]\\,, \\,\\,\\,\n\\hat{\\Pi}(t,x)=\\int \\frac{d k}{i2\\pi}e^{ik x}\\sqrt{\\frac{|k|}{2\\pi K}} \\big[e^{-i v|k| t}\\hat{a}(k)-e^{-i v|k| t}\\hat{a}^\\dagger(-k)\\big]\\, .\n\\end{equation}\nand the Hamiltonian is diagonalized as $\\hat{\\bm H}_0|_{\\eta=0}=\\int dk v|k| \\hat{a}^\\dagger(k)\\hat{a}(k)$. Therefore, the ground state is the vacuum $\\hat{a}(k)|0\\rangle=0$.\nIn the presence of the noise, the fields evolve according to the equation of motion\n\\begin{equation}\n\\partial_t \\hat{\\Pi}=\\frac{1}{\\pi} \\frac{v}{K} \\partial_x^2\\hat{\\phi}-\\frac{\\sqrt{\\gamma}}{\\pi}\\partial_x \\eta_{x}(t)\\,, \\hspace{2pc} \\partial_t\\phi=\\pi Kv \\hat{\\Pi}\\, ,\n\\end{equation}\nwhich are easily solved, leading to the simple result $\n\\langle \\hat{a}^\\dagger(t,k)\\hat{a}(t,q)\\rangle=\\delta(k-q) t\\frac{K\\gamma}{2\\pi} |k| \\hat{F}(k)\n$: the average is taken both with respect to the quantum expectation value and to the statistical fluctuations of the noise.\nThe average mode population can be now fed in the Hamiltonian, resulting in the following heating rate\n\\begin{equation}\n\\frac{\\langle \\bm{H}_0 \\rangle}{L} =e_{\\rm{GS}}+t\\frac{\\kappa_2\\gamma}{\\ell}\\frac{K}{\\pi}v\\, ,\n\\end{equation}\nwhich is in perfect agreement with the short-time expansion of the hydrodynamic prediction \\eqref{eq_short}.\n\n\n\n\\section{The particle-hole form factors at coincident rapidities}\n\n\nWe now show the identity \n\\begin{equation}\\label{Seq_top}\n\\lim_{p_{\\rm p} \\to p_{\\rm h}} \\bra{\\rho} \\bm{O} \\ket{\\rho; \\{p_{\\rm p}, p_{\\rm h}\\}} = V^{\\bf{O}}(p_{\\rm h})=\\frac{1}{2\\pi} \\frac{\\delta\\langle \\bm{O}\\rangle}{\\delta n(p)}\n\\end{equation}\nfor an arbitrary local operator $\\bm O$.\nLet us consider a generic GGE parametrized with a generating function $w(\\lambda)$\n\\begin{equation}\\label{Seq_TBA}\n\\varepsilon(\\lambda)=w(\\lambda)+\\int \\frac{d \\lambda'}{2\\pi} T(\\lambda,\\lambda')\\log(1+ e^{-\\varepsilon(\\lambda')})\\, ,\n\\end{equation}\nwhere the effective energy $\\varepsilon$ parametrizes the filling as $n(\\lambda)=1\/(1+e^{\\varepsilon(\\lambda)})$.\nThen, in Ref. \\cite{10.21468\/SciPostPhys.5.5.054} it has been shown that $V^{\\bm O}$ can be obtained taking the variation of $\\langle \\bm O\\rangle$ with respect to the filling\n\\begin{equation}\\label{Seq_Vo}\n-\\delta\\langle \\bm{O}\\rangle=\\int d\\lambda\\,\\rho(\\lambda)(1-n(\\lambda)) V^{\\bm{O}}(\\lambda) (\\delta w)^\\text{dr}(\\lambda)\\, .\n\\end{equation}\nAbove, we are slightly abusing the notation using the same name for functions defined in the rapidity and momentum space.\nLet us take the variation of Eq. \\eqref{Seq_TBA}\n\\begin{equation}\n\\delta\\varepsilon(\\lambda)=\\delta w(\\lambda)-\\int \\frac{d \\lambda'}{2\\pi} T(\\lambda,\\lambda')n(\\lambda')\\delta \\varepsilon(\\lambda')\\, ,\n\\end{equation}\nwhere we recognize the definition of dressing, i.e. $\\varepsilon(\\lambda)=(\\delta w)^\\text{dr}(\\lambda)$. On the other hand, from the very definition of the filling we have\n$\\delta  n(\\lambda)=-n(\\lambda)(1-n(\\lambda))\\delta \\varepsilon(\\lambda)$. Combining these last two identities in Eq. \\eqref{Seq_Vo} we find\n\\begin{equation}\n\\delta\\langle \\bm{O}\\rangle=\\int d\\lambda\\, \\rho_t(\\lambda) V^{\\bm{O}}(\\lambda) \\delta n(\\lambda)\\, .\n\\end{equation}\nFinally, changing variable from the rapidity to the momentum space $2\\pi \\rho_t(\\lambda) d\\lambda=dp$, Eq. \\eqref{Seq_top} immediately follows.\n\n\n\n\n\n\\end{document}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
{"text":"\\section{Framework for LFV}\n\nWe work within the framework of the Minimal Supersymmetric Standard Model (MSSM) enlarged by three right\nhanded neutrinos and their SUSY partners, where potentially observable \nLFV effects in the charged lepton sector are expected to occur. \nWe further assume a seesaw mechanism\nfor neutrino mass generation and use the parameterisation \n$m_D =\\,Y_\\nu\\,v_2 =\\,\\sqrt {m_N^{\\rm diag}} R \\sqrt {m_\\nu^{\\rm diag}}U^{\\dagger}_{\\rm\nMNS}$, with\n$R$ defined by $\\theta_i$\n($i=1,2,3$);  $v_{1(2)}= \\,v\\,\\cos (\\sin) \\beta$, $v=174$ GeV; \n$m_{\\nu}^\\mathrm{diag}=\\, \\mathrm{diag}\\,(m_{\\nu_1},m_{\\nu_2},m_{\\nu_3})$ denotes the\nthree light neutrino masses, and  \n$m_N^\\mathrm{diag}\\,=\\, \\mathrm{diag}\\,(m_{N_1},m_{N_2},m_{N_3})$ the three heavy\nones. $U_{\\rm MNS}$ is given by\nthe three (light) neutrino mixing angles $\\theta_{12},\\theta_{23}$ and $\\theta_{13}$, \nand three phases, $\\delta, \\phi_1$ and $\\phi_2$. With this \nparameterisation it is easy to accommodate\nthe neutrino data, while leaving room for extra neutrino mixings (from the right\nhanded sector). It further allows for large\nYukawa couplings $Y_\\nu \\sim \\mathcal{O}(1)$ by\nchoosing large entries in $m^{\\rm diag}_N$ and\/or $\\theta_i$. \n\nHere we focus in the particular LFV proccesses: 1) semileptonic $\\tau \\to\n\\mu PP$ ($PP = \\pi^+ \\pi^-, \\pi^0 \\pi^0, K^+ K^-, K^0 \\bar{K}^0$), $\\tau \\to\n\\mu P$ ($P =\n\\pi, \\eta, \\eta^{\\prime}$), $\\tau \\to\n\\mu V$ ($V = \\rho, \\phi$) decays and 2) $\\mu-e$ conversion in heavy nuclei. The predictions \nin the following are for two different constrained MSSM-seesaw scenarios, \nwith universal and non-universal Higgs soft masses. \nThe respective parameters (in addition to the\nprevious neutrino sector parameters) are: \n1) CMSSM-seesaw: $M_0$, $M_{1\/2}$, $A_0$ $\\tan \\beta$, and sign($\\mu$), and \n2) NUHM-seesaw: $M_0$, $M_{1\/2}$, $A_0$ $\\tan \\beta$, sign($\\mu$),\n$M_{H_1}=M_0(1+\\delta_1)^{1\/2}$ and\n$M_{H_2}=M_0(1+\\delta_2)^{1\/2}$. The predictions presented here for\nthe $\\mu-e$ conversion rates include \nthe full set of SUSY one-loop contributing diagrams, mediated by $\\gamma$, Z,\n and Higgs bosons, as well as boxes, and do not use the Leading\n Logarithmic (LLog) nor the mass insertion approximations.\nIn the case of semileptonic tau decays we have not included the boxes\nwhich are clearly subdonimant.\nThe hadronisation of quark bilinears is performed within the\nchiral framework, using Chiral Perturbation Theory and Resonance Chiral\nTheory. This is a very short summary of the works\nin~\\cite{Arganda:2007jw} and~\\cite{Arganda:2008jj} to which we\nrefer the reader for more details.\n\n\\section{Results and discussion}\n\nHere we present the predictions for BR($\\tau \\to \\mu PP$) ($PP = \\pi^+\n\\pi^-, \\pi^0 \\pi^0, K^+ K^-, K^0 \\bar{K}^0$), BR($\\tau \\to \\mu P$) ($P =\n\\pi, \\eta, \\eta^{\\prime}$), BR($\\tau \\to \\mu V$) ($V = \\rho, \\phi$) and\nCR($\\mu-e$, Nuclei) within the previously described framework and\ncompare them with the following experimental bounds: BR$(\\tau \\to \\mu \\pi^+\n\\pi^-) < 4.8 \\times 10^{-7}$, BR$(\\tau \\to \\mu K^+\nK^-) < 8 \\times 10^{-7}$, BR$(\\tau \\to \\mu \\pi) < 5.8 \\times 10^{-8}$,\nBR$(\\tau \\to \\mu \\eta) < 5.1 \\times 10^{-8}$, BR$(\\tau \\to \\mu\n\\eta^{\\prime}) < 5.3 \\times 10^{-8}$, BR$(\\tau \\to \\mu \\rho) < 2\n\\times 10^{-7}$, BR$(\\tau \\to \\mu \\phi) < 1.3 \\times 10^{-7}$,\nCR$(\\mu-e, {\\rm Au}) < 7 \\times 10^{-13}$ and CR$(\\mu-e, {\\rm Ti}) <\n4.3 \\times 10^{-12}$.\n\nAs a general result in LFV processes that can be mediated by Higgs\nbosons we have found that the $H^0$ and $A^0$ contributions are relevant at\nlarge $\\tan\\beta$ if the Higgs masses\nare light enough. It is in this aspect where the main difference between the two\nconsidered scenarios lies. Within the CMSSM, light Higgs $H^0$ and $A^0$\nbosons are only possible for low $M_{\\rm SUSY}$ (here we take $M_{\\rm SUSY} =\nM_0 = M_{1\/2}$ to reduce the number of input parameters). In\ncontrast, within the NUHM, light Higgs bosons can be obtained even at\nlarge $M_{\\rm SUSY}$. In Fig.~\\ref{fig:1} it is shown that some\nspecific choices of $\\delta_1$ and $\\delta_2$ lead to values of\n$m_{H^0}$ and $m_{A^0}$ as low as 110-120 GeV, even for  heavy $M_{\\rm SUSY}$\nvalues above 600 GeV. Therefore, the sensitivity to the Higgs sector\nis higher in the NUHM.\n\n\n\\begin{figure}\n\\begin{tabular}{c}\n\\psfig{file=figs\/mH0_MSUSY.thetai_0.ps,width=50mm,angle=270,clip=}\\\\\n\\psfig{file=figs\/mA0_MSUSY.thetai_neq0.ps,width=50mm,angle=270,clip=}\n\\caption{Predictions of the Higgs boson masses as a function of\n $M_{\\rm SUSY}$ in CMSSM ($\\delta_1 = \\delta_2 = 0$) and NUHM.}\n\\label{fig:1}      \n\\end{tabular}\n\\end{figure}\n\nWe start by presenting the results for the semileptonic tau\ndecays. The mentioned sensitivity to the Higgs sector within the NUHM\nscenario can be seen\nin Fig.~\\ref{fig:2}. Concretely, the BRs of the channels $\\tau \\to \\mu K^+ K^-$,\n$\\tau \\to \\mu K^0 \\bar K^0$, $\\tau \\to \\mu \\pi^0 \\pi^0$, $\\tau\n\\to \\mu \\pi$, $\\tau \\to \\mu \\eta$ and $\\tau \\to \\mu \\eta^{\\prime}$\npresent a growing behaviour with $M_{\\rm SUSY}$, in the large $M_{\\rm\nSUSY}$ region, due to the contribution of light Higgs bosons,\nwhich is non-decoupling. The decays involving Kaons and $\\eta$\nmesons are particularly sensitive to the Higgs contributions because\nof their strange quark content, which has a stronger coupling to the Higgs bosons.\nOn the other hand, the largest predicted\nrates are for $\\tau \\to \\mu \\pi^+ \\pi^-$ and $\\tau \\to \\mu \\rho$,\ndominated by the photon contribution, which are indeed at the present experimental\nreach in the low $M_{\\rm SUSY}$ region.\n\n\\begin{figure}\n\\begin{tabular}{c}\n\\psfig{file=figs\/tau_muPP.MSUSY.ps,width=50mm,angle=270,clip=}\\\\\n\\psfig{file=figs\/tau_muP.MSUSY.ps,width=50mm,angle=270,clip=}\n\\caption{Present sensitivity to LFV in semileptonic $\\tau$ decays\nwithin the NUHM scenario. The horizontal lines denote\nexperimental bounds.}\n\\label{fig:2}      \n\\end{tabular}\n\\end{figure}\n\nThe maximum sensitivity to the Higgs sector is found for $\\tau \\to \\mu\n\\eta$ and $\\tau \\to \\mu \\eta^{\\prime}$ channels, largely dominated by\nthe $A^0$ boson exchange. Fig.~\\ref{fig:3} shows that BR($\\tau \\to \\mu\n\\eta$) reaches the experimental bound for large heaviest neutrino mass,\nlarge $\\tan\\beta$, large $\\theta_i$ angles and low $m_{A^0}$. For the choice of input\nparameters in this figure, it occurs at \n$m_{N_3} = 10^{15}$ GeV, $\\tan\\beta = 60$, $\\theta_2 = 2.9\ne^{i\\pi\/4}$ and $m_{A^0} = 180$ GeV.\nA set of useful formulae for all these channels, within the mass\ninsertion approximation which are\nvalid at large $\\tan\\beta$, are presented in~\\cite{Arganda:2008jj}.\nWe have shown that the predictions with these formulae agree with the\nfull results within a factor of about 2. In the case of $\\tau \\to \\mu\n\\eta$ this comparison is shown in\nFig.~\\ref{fig:3}. Similar conclusions are found for $\\tau \\to \\mu\n\\eta^{\\prime}$. The next relevant channel in sensitivity to the Higgs\nsector is $\\tau \\to \\mu K^+ K^-$, but\nit is still below the present experimental bound. To our knowledge,\nthere are not experimental bounds yet  available for $\\tau \\to \\mu K^0\n\\bar K^0$ and $\\tau \\to \\mu \\pi^0 \\pi^0$.\n\n\\begin{figure}\n\\psfig{file=figs\/tau_mueta.mA0.ps,width=50mm,angle=270,clip=}\n\\caption{Sensitivity to Higgs sector in $\\tau \\to \\mu \\eta$ decays.}\n\\label{fig:3}      \n\\end{figure}\n\nNext we comment on the results for $\\mu-e$ conversion in\nnuclei. Fig.~\\ref{fig:4} shows our predictions of the conversion rates\nfor Titanium as a function of $M_{\\rm SUSY}$ in both CMSSM and NUHM\nscenarios. As in the case of semileptonic tau decays, the sensitivity\nto the Higgs contribution is only manifest in the NUHM scenario. The\npredictions for CR($\\mu-e$, Ti) within the CMSSM scenario are largely\ndominated by the photon contribution and present a decoupling\nbehaviour at large $M_{\\rm SUSY}$. In this case the present\nexperimental bound is only reached at low $M_{\\rm SUSY}$. The\nperspectives for the future are much more promising. If the announced\nsensitivity by PRISM\/PRIME of $10^{-18}$ is finally attained, the full studied\nrange of $M_{\\rm SUSY}$ will be covered. \n\nFig.~\\ref{fig:4} also illustrates that within the NUHM scenario the\nHiggs contribution dominates at large $M_{\\rm SUSY}$ for light Higgs\nbosons. The predicted rates are close to the present experimental bound\nnot only in the low $M_{\\rm SUSY}$ region but also for heavy SUSY\nspectra. As in the previous semileptonic tau decays, we have found in\naddition a simple formula for the conversion rates, within the mass\ninsertion approximation, which is\nvalid at large $\\tan\\beta$~\\cite{Arganda:2007jw} and can be used for\nfurther analysis.\n\n\\begin{figure}\n\\begin{tabular}{cc}\n\\psfig{file=figs\/CRmue_MSUSY.CMSSM.ps,width=50mm,angle=270,clip=}\\\\\n\\psfig{file=figs\/CRmue_MSUSY.NUHM.ps,width=50mm,angle=270,clip=}\n\\caption{Predictions of CR($\\mu-e$, Ti) as a function of\n $M_{\\rm SUSY}$ in the CMSSM (above) and NUHM (below) scenarios.}\n\\label{fig:4}      \n\\end{tabular}\n\\end{figure}\n\nThe predictions of the $\\mu-e$ conversion rates for several nuclei are\ncollected in Fig.~\\ref{fig:5}. We can see again the growing behaviour\nwith $M_{\\rm SUSY}$ in the large $M_{\\rm SUSY}$ region due to the\nnon-decoupling of the Higgs contributions. At present, the most\ncompetitive nucleus for LFV searches is Au where, for the choice of\ninput parameters in this figure, all the predicted rates are above the\nexperimental bound. We have also shown in~\\cite{Arganda:2007jw} that $\\mu-e$ conversion in\nnuclei is extremely sensitive to $\\theta_{13}$, similarly to $\\mu \\to\ne \\gamma$ and $\\mu \\to 3e$ and, therefore, a future\nmeasurement of this mixing angle can help in the searches of LFV in\nthe $\\mu-e$ sector.\n\n\\begin{figure}\n\\psfig{file=figs\/CRmue_MSUSY.nuclei.ps,width=50mm,angle=270,clip=}\n\\caption{Present sensitivity to LFV in $\\mu-e$ conversion for several\nnuclei within NUHM.}\n\\label{fig:5}      \n\\end{figure}\n\nIn conclusion, we have shown that semileptonic tau decays nicely complement\nthe searches for LFV in the $\\tau-\\mu$ sector, in addition to\n$\\tau \\to \\mu \\gamma$. The future prospects for $\\mu-e$ conversion in\nTi are the most promising for LFV searches. Both processes,\nsemileptonic tau decays and $\\mu-e$ conversion in nuclei are indeed more\nsensitive to the Higgs sector than $\\tau \\to 3 \\mu$. \n\n\\begin{theacknowledgments}\nM. Herrero acknowledges the organisers for her\ninvitation to give this talk and for the fruitful conference.   \n\\end{theacknowledgments}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}