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A pair of quasirandom estimations of two-qubit separability probabilities with respect to ten measures -- the Hilbert-Schmidt and nine operator monotone ones

Paul B. Slater

Published Oct 17, 2019• 0 citations
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Abstract

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 $\frac{8}{33} \approx 0.242424$ and $\frac{25}{341} \approx 0.0733138$, respectively, as well as suggestive of the Wigner-Yanase counterpart being $\frac{1}{20}$. However, they appear inconsistent (much too small) with the additional claim that the separability probability associated with the operator monotone (geometric-mean) function $\sqrt{x}$ is $1-\frac{256}{27 \pi ^2} \approx 0.0393251$. But a seeming explanation for this phenomenon is that the volume of states for the $\sqrt{x}$-based measure is infinite, so the validity of the conjecture--as well as an alternative one, $\frac{1}{9} \left(593-60 \pi ^2\right) \approx 0.0915262$, we now introduce--can not be examined through our numerical approach, at least perhaps not without some truncation procedure for extreme values.

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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.
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.
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.
We extend the findings and analyses of our two recent studies (Phys. Rev. A, 75, 032326 [2007] and arXiv:0704.3723) by, first, obtaining numerical estimates of the separability function based on the (Euclidean, flat) Hilbert-Schmidt (HS) metric for the 27-dimensional convex set of quaternionic two-qubit systems. The estimated function appears to be strongly consistent with our previously-formulated Dyson-index (beta = 1, 2, 4) ansatz, dictating that the quaternionic (beta=4) separability function should be exactly proportional to the square of the separability function for the 15-dimensional convex set of two-qubit complex (beta=2) systems, as well as the fourth power of the separability function for the 9-dimensional convex set of two-qubit real (beta=1) systems. In particular, we conclude that S_{quat}(mu) =(6/71)^2 ((3-mu^2) mu)^4 =(S_{complex}(mu))^2. Here, mu =(rho_{11} rho_{44}/(rho_{22} rho_{33})^(1/2), where rho is a 4 x 4 two-qubit density matrix. We can, thus, supplement (and fortify) our previous assertion that the HS separability probability of the two-qubit complex states is 8/33= 0.242424, by claiming that its quaternionic counterpart is 72442944/936239725 = 0.0773765. We also comment on and analyze the odd beta=1 and 3 cases.
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
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 investigate the possibility of deriving analytical formulas for the 15-dimensional separable volumes, in terms of any of a number of metrics of interest (Hilbert-Schmidt [HS], Bures,...), of the two-qubit (four-level) systems. This would appear to require 15-fold symbolic integrations over a complicated convex body (defined by both separability and feasibility constraints). The associated 15-dimensional integrands -- in terms of the Tilma-Byrd-Sudarshan Euler-angle-based parameterization of the 4 x 4 density matrices ρ(math-ph/0202002) -- would be the products of 12-dimensional Haar measure μ_{Haar} (common to each metric) and 3-dimensional measures μ_{metric} (specific to each metric) over the 3d-simplex formed by the four eigenvalues of ρ. We attempt here to estimate/determine the 3-dimensional integrands (the products of the various [known] μ_{metric}'s and an unknown symmetric weighting function W) remaining after the (putative) 12-fold integration of μ_{Haar} over the twelve Euler angles. We do this by fitting W so that the conjectured HS separable volumes and hyperareas (quant-ph/0410238; cf. quant-ph/0609006) are reproduced. We further evaluate a number of possible choices of W by seeing how well they also yield the conjectured separable volumes for the Bures, Kubo-Mori, Wigner-Yanase and (arithmetic) average monotone metrics and the conjectured separable Bures hyperarea (quant-ph/0308037,Table VI). We, in fact, find two such exact (rather similar) choices that give these five conjectured (non-HS) values all within 5%. In addition to the above-mentioned Euler angle parameterization of ρ, we make extensive use of the Bloore parameterization (J. Phys. A 9 [1976], 2059) in a companion set of two-qubit separability analyses.