Das scheint ziemlich bizarr. Ich verwende denselben Befehl Fig.~\ref{fig:twoqubitplot}zweimal in meinem LaTeX-Dokument auf overleaf.com. In der Ausgabe erhalte ich einmal Abb. 2 und einmal Abb. II.
Mico hat nach einem minimalen funktionierenden Beispiel gefragt, aber ich konnte anscheinend nicht ohne weiteres eines bekommen. Also werde ich jetzt mein komplettes Dokument einfügen (eigentlich musste ich es kürzen, um die 30.000-Zeichen-Begrenzung einzuhalten) – bitte kein Plagiat (haha!). Suchen Sie also nach den \ref~{fig:twoqubit}Vorkommen. Ich hoffe, das ist analysierbar – wenn nicht, tut mir das leid.
% ****** Start of file Schack.tex ******
%
%
% This file is part of the APS files in the REVTeX 4 distribution.
% Version 4.0 of REVTeX, August 2001
%
% Copyright (c) 2001 The American Physical Society.
%/Network/Servers/isengard.kitp.ucsb.edu/Volumes/u1/visitors2/slater/MR2443306Cubitt.rtf
% See the REVTeX 4 README file for restrictions and more information.
%\includegraphics[]{../../../../../../../../Network/Servers/buckland.itp.ucsb.edu/Volumes/u1/visitors/slater/Jan2007.pdf}
% TeX'ing this file requires that you have AMS-LaTeX 2.0 installed
% as well as the rest of the prerequisites for REVTeX 4.0
%
% See the REVTeX 4 README file
% It also requires running BibTeX. The commands are as follows:
%ƒ
% 1) latex Schack.tex
% 2) bibtex Schack
% 3) latex Schack.tex
% 4) latex Schack.tex
%
%\documentclass[twocolumn,showpacs,preprintnumbers,amsmath,amssymb]{revtex4}
\documentclass[preprint,showpacs,preprintnumbers,amsmath,amssymb]{revtex4}
\usepackage{amsmath}
\newcommand\numberthis{\addtocounter{equation}{1}\tag{\theequation}}
% Some other (several out of many) possibilities\input{../../../../../../../../Network/Servers/lorien/Volumes/u1/residents/slater/Husimi.tex}
%\documentclass[preprint,aps]{revtex4}
%\documentclass[preprint,aps,draft]{revtex4}
%\documentclass[prb]{revtex4}% Physical Review B
\usepackage{graphicx}% Include figure files
\usepackage{pdfpages}
\usepackage{dcolumn}% Align table columns on decimal point
\usepackage{bm}% bold math
\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{url}
\usepackage{microtype}
\usepackage{graphicx}%
%\nofiles
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{corollary}[theorem]{Corollary}
\newenvironment{proof}[1][Proof]{\begin{trivlist}
\item[\hskip \labelsep {\bfseries #1}]}{\end{trivlist}}
\newenvironment{definition}[1][Definition]{\begin{trivlist}
\item[\hskip \labelsep {\bfseries #1}]}{\end{trivlist}}
\newenvironment{example}[1][Example]{\begin{trivlist}
\item[\hskip \labelsep {\bfseries #1}]}{\end{trivlist}}
\newenvironment{remark}[1][Remark]{\begin{trivlist}
\item[\hskip \labelsep {\bfseries #1}]}{\end{trivlist}}
\newcommand{\qed}{\nobreak \ifvmode \relax \else
\ifdim\lastskip<1.5em \hskip-\lastskip
\hskip1.5em plus0em minus0.5em \fi \nobreak
\vrule height0.75em width0.5em depth0.25em\fi}
\newcommand{\norm}[1]{\left\lVert#1\right\rVert}
\renewcommand{\vec}[1]{\mathbf{#1}}
\begin{document}
\preprint{}
\title{Quasirandom Estimation of Bures and Hilbert-Schmidt Separability Probabilities and Associated Rational-Valued Conjectures}
\author{Paul B. Slater}
\email{[email protected]}
\affiliation{%
Kavli Institute for Theoretical Physics, University of California, Santa Barbara, CA 93106-4030\\
}
\date{\today}% It is always \today, today,
% but any date may be explicitly specified
\begin{abstract}
We employ a quasirandom methodology, recently developed by Martin Roberts, to estimate the separability probabilities, with respect to the Bures (minimal monotone/statistical distinguishability) measure, of generic two-qubit and two-rebit states. This procedure, based on generalized properties of the golden ratio,
yielded, in the course of almost seventeen billion iterations (recorded at intervals of five million), two-qubit estimates
repeatedly close to nine decimal places to
$\frac{25}{341} =\frac{5^2}{11 \cdot 31} \approx 0.073313783$. The two-qubit probabilities based, alternatively, on the Hilbert-Schmidt and operator monotone function $\sqrt{x}$ measures are (still subject to formal proof) essentially known to be $\frac{8}{33} = \frac{2^3}{3 \cdot 11}$ and $1-\frac{256}{27 \pi^2}=1-\frac{4^4}{3^3 \pi^2}$, respectively.
Further, Lovas and Andai have proven that the corresponding pair of two-rebit probabilities is $\frac{29}{64} = \frac{29}{2^6}$ and approximately 0.26223. In the Bures two-rebit case, we do not presently perceive an exact value (rational or otherwise) for an estimate of 0.15709623, based on over twenty-three billion
iterations. We re-examine, strongly supporting, conjectures that the Hilbert-Schmidt qubit-{\it qutrit} and rebit-{\it retrit} separability probabilities are
$\frac{27}{1000}=\frac{3^3}{2^3 \cdot 5^3}$ and $\frac{860}{6561}= \frac{2^2 \cdot 5 \cdot 43}{3^8}$, respectively.
The Bures qubit-qutrit case--for which Khvedelidze and Rogojin gave an estimate of 0.0014--is analyzed too, with quasirandom sequences of dimension 144.
\end{abstract}
\pacs{Valid PACS 03.67.Mn, 02.50.Cw, 02.40.Ft, 02.10.Yn, 03.65.-w}
% Classification Scheme.
\keywords{separability probabilities, two-qubits, two-rebits, Hilbert-Schmidt measure, random matrix theory, quaternions, PPT-probabilities, operator monotone functions, Bures measure, Lovas-Andai functions, quasirandom sequences, golden ratio, qubit-qutrit, rebit-retrit, inverse normal cumulative distribution}
\maketitle
\section{Introduction}
It has now been formally proven by Lovas and Andai \cite[Thm. 2]{lovas2017invariance} that the separability probability with respect to Hilbert-Schmidt (flat/Euclidean/Frobenius) measure \cite{zyczkowski2003hilbert} \cite[sec. 13.3]{bengtsson2017geometry} of the 9-dimensional convex set of
two-rebit states \cite{Caves2001} is $\frac{29}{64}=\frac{29}{2^6}$. Additionally, the multifaceted evidence \cite{slater2017master,khvedelidze2018generation,milz2014volumes,fei2016numerical,shang2015monte,slater2013concise,slater2012moment,slater2007dyson}--including a recent ``master'' extension \cite{slater2017master,slater2018extensions} of the Lovas-Andai framework to {\it generalized} two-qubit states--is strongly compelling that the corresponding value for the 15-dimensional convex set of two-qubit states is $\frac{8}{33}=\frac{2^3}{3 \cdot 11}$ (with that of the 27-dimensional convex set of two-quater[nionic]bits being $\frac{26}{323}=\frac{2 \cdot 13}{17 \cdot 19}$ [cf. \cite{adler1995quaternionic}], among other still higher-dimensional companion random-matrix related results). (Certainly, one can, however, still aspire to a yet greater
``intuitive'' understanding of these assertions, particularly in some ``geometric/visual'' sense [cf. \cite{szarek2006structure,samuel2018lorentzian,avron2009entanglement,braga2010geometrical,gamel2016entangled,jevtic2014quantum}], as well as further formalized proofs.)
Additionally, appealing hypotheses parallel to these rational-valued results have been advanced--based on extensive sampling--that the Hilbert-Schmidt separability probabilities for the 35-dimensional qubit-{\it qutrit} and 20-dimensional rebit-{\it retrit} states are
$\frac{27}{1000}=\frac{3^3}{2^3 \cdot 5^3}$ and $\frac{860}{6561}= \frac{2^2 \cdot 5 \cdot 43}{3^8}$, respectively \cite[eqs. (15),(20)]{slater2018extensions} \cite[eq. (33)]{milz2014volumes}. (These will be further examined in sec.~\ref{HSsection} below.)
It is of interest to compare/contrast these finite-dimensional studies with those other quantum-information-theoretic ones, presented in the recent comprehensive volume of Aubrun and Szarek
\cite{aubrun2017alice}, employing the quite different concepts of {\it asymptotic geometric analysis}.
By a separability probability, we mean the ratio of the volume of the separable states to the volume of all (separable and entangled) states, as proposed, apparently first, by {\.Z}yczkowski, Horodecki, Sanpera and Lewenstein \cite{zyczkowski1998volume} (cf. \cite{petz1996geometries,e20020146,singh2014relative,batle2014geometric}). The present author was, then, led--pursuing an interest in ``Bayesian quantum mechanics" \cite{slater1994bayesian,slater1995quantum} and the concept of a ``quantum Jeffreys prior" \cite{kwek1999quantum}--to investigate how such separability probabilities might depend upon the choice of various possible measures on the quantum states \cite{petz1996geometries}.
Of particular initial interest was the the Bures/statistical distinguishability (minimal monotone) measure \cite{slater2000exact,sarkar2019bures,vsafranek2017discontinuities,forrester2016relating, braunstein1994statistical}. (``The Bures metric
plays a distinguished role since it is the only metric which is also monotone, Fisher- adjusted, Fubini-Study-adjusted, and Riemannian" \cite{forrester2016relating}. Bej and Deb have recently ``shown that if a qubit gets entangled with another ancillary qubit then negativity, up to a constant factor, is equal to square root of a specific Riemannian metric defined on the metric space corresponding to the state space of the qubit" \cite{bej2018geometry}.)
In \cite[sec. VII.C]{slater2017master}, we recently reported, building upon analyses of Lovas and Andai \cite[sec. 4]{lovas2017invariance}, a two-qubit separability probability equal to $1 -\frac{256}{27 \pi^2} =1- \frac{2^8}{3^3 \pi^2} \approx 0.0393251$. This was based on another
(of the infinite family of) operator monotone functions, namely
$\sqrt{x}$. (The Bures measure itself is associated with the operator monotone function $\frac{1+x}{2}$.) (Let us note that the complementary ``entanglement probability'' is simply $\frac{256}{27 \pi^2} \approx 0.960675$. There appears to be no intrinsic reason
to prefer/privilege one of these two forms (separability, entanglement) of probability to the other [cf. \cite{dunkl2015separability}]. We observe that the variable denoted $K_s = \frac{(s+1)^{s+1}}{s^s}$, equalling $\frac{256}{27} = \frac{4^4}{3^3}$, for $s=3$, is frequently employed as an upper limit of integration in the Penson-{\.Z}yczkowski paper, ``Product of Ginibre matrices: Fuss-Catalan and Raney
distributions'' \cite[eqs. (2), (3)]{penson2011product}.)
Interestingly, Lovas and Andai ``argue that from the separability probability point of view, the main difference between the Hilbert-Schmidt measure and the volume form
generated by the operator monotone function $x \rightarrow \sqrt{x}$ is a special distribution on the unit ball in operator norm of
$2 \times 2$ matrices, more precisely in the Hilbert-Schmidt case one faces a uniform distribution on the whole unit ball and for
monotone volume forms one obtains uniform distribution on the surface of the unit ball'' \cite[p. 2]{lovas2017invariance}.
A formula $Q(k,\alpha)= G_1^k(\alpha) G_2^k(\alpha)$, for $k = -1, 0, 1,\ldots 9$ was given in \cite[p. 26]{slater2016formulas}. It yields that part of the {\it total} separability probability, $P(k,\alpha)$,
for generalized (real [$d=1$], complex [$d=2$], quaternionic [$d=4$],\ldots) two-qubit states endowed with random induced measure, for which the determinantal inequality $|\rho^{PT}| >|\rho|$ holds. Here $\rho$ denotes a $4 \times 4$ density matrix, obtained by tracing over the pure states in $4 \times (4 +k)$-dimensions, and $\rho^{PT}$, its partial transpose. Further, $\alpha$ is a Dyson-index-like parameter with $\alpha = 1$ for the standard (15-dimensional) convex set of (complex) two-qubit states.
Further, in the specific Hilbert-Schmidt case ($k=0$), we can apparently employ \cite[p. 26]{slater2016formulas}
\begin{equation} \label{InducedMeasureCase}
\mathcal{P}_{sep/PPT}(0,d)= 2 Q(0,d)= 1-
\frac{\sqrt{\pi } 2^{-\frac{9 d}{2}-\frac{5}{2}} \Gamma \left(\frac{3 (d+1)}{2}\right)
\Gamma \left(\frac{5 d}{4}+\frac{19}{8}\right) \Gamma (2 d+2) \Gamma \left(\frac{5
d}{2}+2\right)}{\Gamma (d)} \times
\end{equation}
\begin{displaymath}
\, _6\tilde{F}_5\left(1,d+\frac{3}{2},\frac{5 d}{4}+1,\frac{1}{4} (5 d+6),\frac{5
d}{4}+\frac{19}{8},\frac{3 (d+1)}{2};\frac{d+4}{2},\frac{5
d}{4}+\frac{11}{8},\frac{1}{4} (5 d+7),\frac{1}{4} (5 d+9),2 (d+1);1\right).
\end{displaymath}
That is, for $k=0$, we obtain the previously reported Hilbert-Schmidt formulas, with
(the real case) $Q(0,1) = \frac{29}{128}$, (the standard complex case) $Q(0,2)=\frac{4}{33}$, and
(the quaternionic case) $Q(0,4)= \frac{13}{323}$---the three simply
equalling $ P(0,\alpha)/2$.
More generally, $Q(k,d)$ gives that portion, for random {\it induced} measure, parameterized by $k$, of the total separability/PPT-probability for which the determinantal inequality
$|\rho^{PT}| >|\rho|$ holds \cite[eq. (84)]{slater2017master}. (The [Dyson-index] symbol $d$ here is distinct from the use of it below as the length of a sequence in the quasirandom procedure.)
\section{Application of Quasirandom Methodology}
We examine the question of whether Bures two-qubit and two-rebit separability probability estimation can be accelerated--with superior convergence properties--by, rather than using, as typically done,
independently-generated normal variates for the Ginibre ensembles at each iteration, making use of normal variates {\it jointly} generated by employing low-discrepancy (quasi-Monte Carlo) sequences \cite{leobacher2014introduction}. In particular, we have employed an ``open-ended'' sequence (based on extensions of the golden ratio \cite{livio2008golden}) recently introduced by Martin Roberts in the detailed ``blog post", ``The Unreasonable Effectiveness
of Quasirandom Sequences'' \cite{Roberts}.
Roberts notes: ``The solution to the
$d$-dimensional problem, depends on a special constant $\phi_d$, where $\phi_d$ is the value of the smallest, positive real-value of x such that''
\begin{equation}
x^{d+1}=x+1,
\end{equation}
($d=1$, yielding the golden ratio, and $d=2$, the ``plastic constant'' \cite{Roberts32D}).
The $n$-th terms in the quasirandom (Korobov) sequence take the form
\begin{equation} \label{QR}
(\alpha _0+n \vec{\alpha}) \bmod 1, n = 1, 2, 3, \ldots
\end{equation}
where we have the $d$-dimensional vector,
\begin{equation} \label{quasirandompoints}
\vec{\alpha} =(\frac{1}{\phi_d},\frac{1}{\phi_d^2},\frac{1}{\phi_d^3},\ldots,\frac{1}{\phi_d^d}). "
\end{equation}
The additive constant $\alpha_0$ is typically taken to be 0. ``However, there are some arguments, relating to symmetry, that suggest that $\alpha_0=\frac{1}{2}$
is a better choice,'' Roberts observes.
These points (\ref{quasirandompoints}), lying in the $d$-dimensional hypercube $[0,1]^d$, can be converted to (quasirandomly distributed) normal variates, required for implementation of the Osipov-Sommers-{\.Z}yczkowski formula (\ref{JointBuresHSformula}), using the inverse of the cumulative distribution function \cite[Chap. 2]{devroye1986}.
Impressively, in this regard, Henrik Schumacher developed for us a specialized algorithm that accelerated the default Mathematica command InverseCDF for the normal distribution approximately {\it ten-fold}, as reported in the highly-discussed post \cite{Schumacher}--allowing us to vastly increase the realization rate.
We take $d=36$ and 64 in the Roberts methodology, using the Osipov-Sommers-{\.Z}yczkowski (real and complex) interpolation formulas to estimate the Bures two-rebit and two-qubit separability probabilities, respectively. In the two-qubit case, 32 of the 64 variates are used in generating the Ginibre matrix $A$, and the other 32, for the unitary matrix $U$.
(A subsidiary question--which appeared in the discussion with Roberts \cite{Roberts32D}--is the relative effectiveness of employing--to avoid possible ``correlation'' effects--the {\it same} 32-dimensional sequence but at different $n$'s for $A$ and $U$, rather than a single 64-dimensional one, as pursued here. A small analysis of ours in this regard did not indicate this to be a meritorious approach.) In the two-rebit case, 20 variates are used to generate
the $4 \times 5$ matrix A, and the other 16 for an orthogonal $4 \times 4$ matrix $O$.
It would be of substantial interest to compare/contrast the relative merits of the estimation of this pair of Bures separability probabilities in the present 36- and 64-dimensional settings with earlier studies
(largely involving Euler-angle parameterizations of $4 \times 4$ density matrices \cite{tilma2002parametrization}), in which 9- and 15-dimensional integration problems were addressed \cite{slater2005silver,slater2009eigenvalues} (cf. \cite{maziero2015random}). In the higher-dimensional frameworks used here, the integrands are effectively unity, with each randomly generated matrix being effectively assigned equal weight, while not so in the other cases. In \cite{ExperimentalData}, we asked the question ``Can `experimental data from a quantum computer' be used to test separability probability conjectures?'', following the analyses of Smart, Schuster and Mazziotti in their article \cite{ssm}, ``Experimental data from a quantum computer verifies the generalized Pauli exclusion principle'', in which
``quantum many-fermion states are randomly prepared on the quantum computer and tested for constraint violations''.
Using the indicated, possibly superior parameter value
$\alpha_0= \frac{1}{2}$ in (\ref{QR}), this quasirandom/normal-variate-generation procedure has so far
yielded a two-qubit estimate, based on 16,895,000,000 iterations, of 0.073313759. This is closely fitted by the two (themselves very near) values
$\frac{25}{341} =\frac{5^2}{11 \cdot 31} \approx 0.07331378299$ and (as suggested by the WolframAlpha.com site) $\frac{\sqrt{51}}{\pi ^4} \approx 0.07331377752$. (Informally, Charles Dunkl wrote: "I would hate to think that the answer is $\frac{\sqrt{51}}{\pi^4}$- that is just ugly. One hopes for a rational number.") At 10,850,000,000 iterations, interestingly, the estimate of 0.0733137814 matched $\frac{25}{341}$ to nearly eight decimal places. The estimate of 0.0733137847 obtained at the considerably smaller number of iterations of 1,445,000,000, was essentially as close too.
The Hilbert-Schmidt measure counterpart is (still subject to formal proof) essentially known to be $\frac{8}{33} = \frac{2^3}{3 \cdot 11}$ \cite{slater2017master,khvedelidze2018generation,milz2014volumes,fei2016numerical,shang2015monte,slater2013concise,slater2012moment,slater2007dyson}.
In the two-rebit instance, we obtained a Bures estimate, based on 23,460,000,000
iterations, of 0.157096234. (This is presumably, at least as accurate as the considerably, just noted, smaller sample based two-qubit estimate--apparently corresponding to $\frac{25}{341}$. Nevertheless, we do not presently perceive any possible exact--rational or otherwise--fits to this estimate.) The Hilbert-Schmidt two-rebit separability probability
has been proven by Lovas and Andai to be $\frac{29}{64} = \frac{29}{2^6}$ \cite[Thm. 2]{lovas2017invariance}.
In Figs.~\ref{fig:twoqubitplot} and \ref{fig:tworebitplot} we show the development of the Bures separability probability estimation procedure in the two cases at hand. (Much earlier versions of these [$\alpha_0=\frac{1}{2}$] plots--together with [less intensive] estimates using $\alpha_0=0$--were displayed as Figs. 5 and 6 in \cite{slater2018extensions}.)
\begin{figure}
\includegraphics[]{BuresQubit2.pdf}
\label{fig:twoqubitplot}
\caption{Two-qubit Bures separability probability estimates--divided by $\frac{25}{341}$--as a function of the number of iterations of the quasirandom procedure, using $\alpha_0=\frac{1}{2}$. This ratio is equal to 1 to nearly eight decimal places at: 1,445,000,000; 10,850,000,000; 11,500,000,000; and 16,075,000,000 iterations. Estimates are recorded at intervals of five million iterations.}
\end{figure}
\begin{figure}
\includegraphics[]{BuresRebit2.pdf}
\label{fig:tworebitplot}
\caption{Two-rebit Bures separability probability estimates as a function of the number of iterations of the quasirandom procedure, using $\alpha_0=\frac{1}{2}$. Estimates are recorded at intervals of five million iterations.}
\end{figure}
\section{Examination of Hilbert-Schmidt Qubit-Qutrit and Rebit-Retrit Separability Conjectures} \label{HSsection}
Based on extensive (standard) random sampling of independent normal variates, in \cite[eqs. (15),(20)]{slater2018extensions}, we have conjectured that the Hilbert-Schmidt separability probabilities for the 35-dimensional qubit-{\it qutrit} and 20-dimensional rebit-{\it retrit} states are (also interestingly rational-valued)
$\frac{27}{1000}=\frac{3^3}{2^3 \cdot 5^3} =0.027$ and $\frac{860}{6561}= \frac{2^2 \cdot 5 \cdot 43}{3^8} \approx 0.1310775796$, respectively .
In particular, on the basis of 2,900,000,000 randomly-generated
qubit-qutrit density matrices, an estimate (with 78,293,301
separable density matrices found) was obtained, yielding an associated separability probability of 0.026997690. (Milz and Strunz had given a confidence interval of $0.02700 \pm 0.00016$ for this probability \cite[eq. (33)]{milz2014volumes}, while Khvedelidze and Rogojin reported an estimate of 0.0270 \cite[Tab. 1]{khvedelidze2018generation}--but also only 0.0014 for the Bures counterpart [sec.~\ref{BuresQubitQutrit}].)
Further, on the basis of 3,530,000,000 randomly-generated
rebit-retrit density matrices, with respect to Hilbert-Schmidt measure, an estimate (with 462,704,503
separable density matrices found) was obtained for an associated separability probability of 0.1310777629. The associated
$95\%$ confidence interval is $[0.131067, 0.131089]$.
Applying the quasirandom methodology here to further appraise this pair of conjectures, we obtain Figs.~\ref{fig:QuasiRandomQubitQutrit} and \ref{fig:QuasiRandomRebitRetrit}. (We take the dimensions $d$ of the sequences of normal variates generated to be 72 and 42, respectively.)
\begin{figure}
\centering
\includegraphics{QuasiRandomQubitQutrit.pdf}
\caption{Qubit-qutrit Hilbert-Schmidt separability probability estimates--divided by $\frac{27}{1000}$--as a function of the number of iterations of the quasirandom procedure, using $\alpha_0=\frac{1}{2}$. Estimates are recorded at intervals of five million iterations.}
\label{fig:QuasiRandomQubitQutrit}
\end{figure}
\begin{figure}
\centering
\includegraphics{QuasiRandomRebitRetrit.pdf}
\caption{Rebit-retrit Hilbert-Schmidt separability probability estimates--divided by $\frac{860}{6561} = \frac{2^2 \cdot 5 \cdot 43}{3^8}$--as a function of the number of iterations of the quasirandom procedure, using $\alpha_0=\frac{1}{2}$. Estimates are recorded at intervals of five million iterations.}
\label{fig:QuasiRandomRebitRetrit}
\end{figure}
Interestingly, as in Fig.~\ref{fig:twoqubitplot}, we observe some
drift away--with increasing iterations-- from early particularly close fits
to the two conjectures. But, as in Fig.~\ref{fig:twoqubitplot}--assuming the validity
of the conjectures--we might anticipate the estimates re-approaching more closely
the conjectured values. It would seem that any presumed eventual convergence is not simply a
straightforward monotonic process--perhaps somewhat comprehensible in view of the very high dimensionalities (72, 42) of the sequences involved. (The last recorded separability probabilities--in these ongoing analyses--were 0.0269923 and 0.1310848, based on 1,850,000,000 and
2,415,000,000 iterations, respectively.)
In \cite[App. B]{slater2017master}, we reported an effort to extend the innovative framework of Lovas and Andai \cite{lovas2017invariance} to such qubit-qutrit and rebit-retrit settings.
\section{Bures Qubit-Qutrit Analysis} \label{BuresQubitQutrit}
In Table 1 of their recent study, "On the generation of random ensembles of qubits and qutrits: Computing separability probabilities for fixed rank states'' \cite{khvedelidze2018generation}, Khvedelidze and Rogojin report an estimate (no sample size being given) of 0.0014 for the separability probability of the 35-dimensional convex set of qubit-qutrit states. We undertook a study of this issue, once again employing the quasirandom methodology advanced by Roberts (with $d$ now equal to $144=2 \cdot 72$), in implementing the Osipov-Sommers-{\.Z}yczkowski formula (\ref{JointBuresHSformula}) given above with $x=\frac{1}{2}$. (For the companion Bures rebit-retrit estimation, we would have a smaller $d$, that is, 78--but, given our Bures two-rebit analysis above, we were not optimistic in perceiving an exact value.) In Fig.~\ref{fig:BuresQubitQutrit} we show a plot of our corresponding computations. The estimates--recorded at intervals of one million--are in general agreement with the reported value of Khvedelidze and Rogojin. The last value (after 813 million iterations) was $\frac{227699}{162800000}= \approx 0.0013986425$. This can be fitted by $\frac{1}{715} =\frac{1}{5 \cdot 11 \cdot 13} \approx 0.0013986$.
\begin{figure}
\centering
\includegraphics{BuresQubitQutrit.pdf}
\caption{Qubit-qutrit Bures separability probability estimates. Khvedelidze and Rogojin reported an estimate of 0.0014.}
\label{fig:BuresQubitQutrit}
\end{figure}
\section{Concluding Remarks}
We should stress that the problem of {\it formally deriving} the Bures two-rebit and two-qubit separability probabilities, and, thus, testing the candidate value ($\frac{25}{341}$) advanced here, certainly currently seems intractable--even, it would seem, in the pioneering framework of Lovas and Andai \cite{lovas2017invariance}. (Perhaps some formal advances can be made, in such regards, with respect to $X$-states [cf. \cite{xiong2017geometric}].)
\begin{acknowledgements}
This research was supported by the National Science Foundation under Grant No. NSF PHY-1748958.
\end{acknowledgements}
\bibliography{main}% Produces the bibliography via BibTeX.
\end{document}
%
% ****** End of file Cupdate.tex ******
Antwort1
Verschieben Sie alle \label{}Befehle nach \caption{}den Befehlen. Beispiel:
\begin{figure}\centering
\includegraphics{BuresRebit2.pdf}
\caption{Two-rebit Bures separability probability estimates as a function of the number of iterations of the quasirandom procedure, using $\alpha_0=\frac{1}{2}$. Estimates are recorded at intervals of five million iterations.}
\label{fig:tworebitplot}
\end{figure}
Antwort2
Dies scheint an der Reihenfolge von captionund labelin zwei Ihrer Abbildungen zu liegen. In beiden Fällen steht fig:twoqubitplotund vor fig:tworebitplotdem , was dazu führt, dass auf Abschnitt (II) und nicht auf Abbildung (2 und 3) verwiesen wird.labelcaption