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arXiv:2101.07716

Hilbert-Schmidt Separability Probabilities from Bures Ensembles and vice versa: Applications to Quantum Steering Ellipsoids and Monotone Metrics

Paul B. Slater

Published Jan 19, 2021• 0 citations
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Abstract

We reexamine a recent analysis in which, using the volume of the associated quantum steering ellipsoid (QES) as a measure, we sought to estimate the probability that a two-qubit state is separable. In the estimation process, we, in effect, sought to attach to states random with respect to Hilbert-Schmidt (HS) measure, the corresponding QES volumes. However, a study of the relations between HS and Bures ensembles and their well-supported separability probabilities of $\frac{8}{33}$ and $\frac{25}{341}$, respectively, now lead us to explore as a possible alternative measure, the QES volume divided by the $Π_{j

DOI
10.48550/arxiv.2101.07716

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We reexamine a recent analysis in which, using the volume of the associated quantum steering ellipsoid (QES) as a measure, we sought to estimate the probability that a two-qubit state is separable. In the estimation process, we, in effect, sought to attach to states random with respect to Hilbert-Schmidt (HS) measure, the corresponding QES volumes. However, a study of the relations between HS and Bures ensembles and their well-supported separability probabilities of 833\frac{8}{33}338​ and 25341\frac{25}{341}34125​, respectively, now lead us to explore as a possible alternative measure, the QES volume divided by the $Π_{j
Employing the volume of quantum steering ellipsoids (QSEs) as a measure on the fifteen-dimensional convex set of two-qubit states, we estimate the ratio of the integral of the measure over the separable states to its integral over all (separable and entangled) states to be 0.0288. This can be contrasted with the considerably larger separability ratios (probabilities) of 833=233⋅11≈0.242424\frac{8}{33} = \frac{2^3}{3 \cdot 11} \approx 0.242424338​=3⋅1123​≈0.242424 and 25341=5211⋅31≈0.0733138\frac{25}{341}=\frac{5^2}{11 \cdot 31} \approx 0.073313834125​=11⋅3152​≈0.0733138 that various forms of evidence point to with the use of the prominent Hilbert-Schmidt and Bures measures, respectively. The questions of whether the ratio in the QSE setting can be more precisely obtained or even exactly computed, as well as whether a metric can be constructed, the volume element of which yields the measure, remain to be addressed. We also investigate related issues pertaining to absolute separability. Further, we examine the behavior of the separability probability--constant in the Hilbert-Schmidt case and decreasing in the Bures--as a function of the Bloch vector norm in the QSE instance. It appears to increase approaching the pure state boundary.
We conduct a pair of quasirandom estimations of the separability probabilities with respect to ten measures on the 15-dimensional convex set of two-qubit states, using its Euler-angle parameterization. The measures include the (non-monotone) Hilbert-Schmidt one, plus nine based on operator monotone functions. Our results are supportive of previous assertions that the Hilbert-Schmidt and Bures (minimal monotone) separability probabilities are 833≈0.242424\frac{8}{33} \approx 0.242424338​≈0.242424 and 25341≈0.0733138\frac{25}{341} \approx 0.073313834125​≈0.0733138, respectively, as well as suggestive of the Wigner-Yanase counterpart being 120\frac{1}{20}201​. However, they appear inconsistent (much too small) with the additional claim that the separability probability associated with the operator monotone (geometric-mean) function x\sqrt{x}x​ is 1−25627π2≈0.03932511-\frac{256}{27 \pi ^2} \approx 0.03932511−27π2256​≈0.0393251. But a seeming explanation for this phenomenon is that the volume of states for the x\sqrt{x}x​-based measure is infinite, so the validity of the conjecture--as well as an alternative one, 19(593−60π2)≈0.0915262\frac{1}{9} \left(593-60 \pi ^2\right) \approx 0.091526291​(593−60π2)≈0.0915262, we now introduce--can not be examined through our numerical approach, at least perhaps not without some truncation procedure for extreme values.
Paralleling our recent computationally intensive (quasi-Monte Carlo) work for the case N=4N=4N=4 (e-print quant-ph/0308037), we undertake the task for N=6N=6N=6 of computing to high numerical accuracy, the formulas of Sommers and \ifmmode \dot{Z}\else \.{Z}\fi{}yczkowski (e-print quant-ph/0304041) for the $({N}^{2}\ensuremath{-}1)$-dimensional volume and $({N}^{2}\ensuremath{-}2)$-dimensional hyperarea of the (separable and nonseparable) $N\ifmmode\times\else\texttimes\fi{}N$ density matrices, based on the Bures (minimal monotone) metric---and also their analogous formulas (e-print quant-ph/0302197) for the (nonmonotone) flat Hilbert-Schmidt metric. With the same seven 109{10}^{9}109 well-distributed (``low-discrepancy'') sample points, we estimate the unknown volumes and hyperareas based on five additional (monotone) metrics of interest, including the Kubo-Mori and Wigner-Yanase. Further, we estimate all of these seven volume and seven hyperarea (unknown) quantities when restricted to the separable density matrices. The ratios of separable volumes (hyperareas) to separable plus nonseparable volumes (hyperareas) yield estimates of the separability probabilities of generically rank-6 (rank-5) density matrices. The (rank-6) separability probabilities obtained based on the 35-dimensional volumes appear to be---independently of the metric (each of the seven inducing Haar measure) employed---twice as large as those (rank-5 ones) based on the 34-dimensional hyperareas. (An additional estimate---33.9982---of the ratio of the rank-6 Hilbert-Schmidt separability probability to the rank-4 one is quite clearly close to integral too.) The doubling relationship also appears to hold for the N=4N=4N=4 case for the Hilbert-Schmidt metric, but not the others. We fit simple exact formulas to our estimates of the Hilbert-Schmidt separable volumes and hyperareas in both the N=4N=4N=4 and N=6N=6N=6 cases.
We conduct a pair of quasirandom estimations of the separability probabilities with respect to 10 measures on the 15-dimensional convex set of two-qubit states, using its Euler-angle parametrization. The measures include the (nonmonotone) Hilbert–Schmidt one, plus nine others based on operator monotone functions. Our results are supportive of previous assertions that the Hilbert–Schmidt and Bures (minimal monotone) separability probabilities are [Formula: see text] and [Formula: see text], respectively, as well as suggestive of the Wigner–Yanase counterpart being [Formula: see text]. However, one result appears inconsistent (much too small) with an earlier claim of ours that the separability probability associated with the operator monotone (geometric-mean) function [Formula: see text] is [Formula: see text]. But a seeming explanation for this disparity is that the volume of states for the [Formula: see text]-based measure is infinite. So, the validity of the earlier conjecture — as well as an alternative one, [Formula: see text], we now introduce — cannot be examined through the numerical approach adopted, at least perhaps not without some truncation procedure for extreme values.
Due to recent important work of \ifmmode \dot{Z}\else \.{Z}\fi{}yczkowski and Sommers [J. Phys. A 36, 10115 (2003); 36, 10083 (2003)], exact formulas are available (in terms of both the Hilbert-Schmidt and the Bures metrics) for the $({n}^{2}\ensuremath{-}1)$-dimensional and $[n(n\ensuremath{-}1)∕2\ensuremath{-}1]$-dimensional volumes of the complex and real $n\ifmmode\times\else\texttimes\fi{}n$ density matrices. However, no comparable formulas are available for the volumes (and, hence, probabilities) of various separable subsets of them. We seek to clarify this situation for the Hilbert-Schmidt metric for the simplest possible case of n=4n=4n=4, that is, two-qubit systems. Making use of the density matrix $(\ensuremath{\rho})$ parametrization of Bloore [J. Phys. A 9, 2059 (1976)], we are able to reduce each of the real and complex volume problems to the calculation of a one-dimensional integral, the single relevant variable being a certain ratio of diagonal entries, $\ensuremath{\nu}={\ensuremath{\rho}}_{11}{\ensuremath{\rho}}_{44}∕{\ensuremath{\rho}}_{22}{\ensuremath{\rho}}_{33}$. The associated integrand in each case is the product of a known Jacobian (highly oscillatory near $\ensuremath{\nu}=1$) and a certain unknown univariate function, which our extensive numerical (quasi Monte Carlo) computations indicate is very closely proportional to an (incomplete) Beta function ${B}_{\ensuremath{\nu}}(a,b)$, with a=1∕2a=1∕2a=1∕2, b=3b=\sqrt{3}b=3​ in the real case, and a=26∕5,b=3∕2a=2\sqrt{6}∕5,b=3∕\sqrt{2}a=26​∕5,b=3∕2​ in the complex case. Assuming the full applicability of these specific incomplete Beta functions, we undertake separable-volume calculations.
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 fifteen billion iterations, two-qubit estimates repeatedly agreeing to close to nine decimal places with 25341=5211⋅31≈0.07331378299\frac{25}{341} =\frac{5^2}{11 \cdot 31} \approx 0.0733137829934125​=11⋅3152​≈0.07331378299. The corresponding probabilities based on the Hilbert-Schmidt and operator monotone function x\sqrt{x}x​ measures are (still subject to formal proof) essentially known to be 833=233⋅11\frac{8}{33} = \frac{2^3}{3 \cdot 11}338​=3⋅1123​ and 1−25627π2=1−4433π21-\frac{256}{27 \pi^2}=1-\frac{4^4}{3^3 \pi^2}1−27π2256​=1−33π244​, respectively. Further, the analogous pair of two-rebit probabilities has been proven by Lovas and Andai to be 2964=2926\frac{29}{64} = \frac{29}{2^6}6429​=2629​ and approximately 0.26223. In the Bures two-rebit case, we do not presently perceive an exact value corresponding to our quasirandom estimate of 0.15709715, based on over twenty billion iterations. The quasirandom methodology can also be applied to test recent conjectures that the Hilbert-Schmidt qubit-qutrit and rebit-retrit separability probabilities are 271000=3323⋅53\frac{27}{1000}=\frac{3^3}{2^3 \cdot 5^3}100027​=23⋅5333​ and 8606561=22⋅5⋅4338\frac{860}{6561}= \frac{2^2 \cdot 5 \cdot 43}{3^8}6561860​=3822⋅5⋅43​, respectively.