Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:quant-ph/0410238 · Physical Review A

Qubit-qutrit separability-probability ratios

Paul B. Slater

Published May 16, 2005• 29 citations
arXivHTMLPDF

Abstract

Paralleling our recent computationally intensive (quasi-Monte Carlo) work for the case $N=4$ (e-print quant-ph/0308037), we undertake the task for $N=6$ 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 ${10}^{9}$ 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 $N=4$ 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 $N=4$ and $N=6$ cases.

DOI
10.1103/physreva.71.052319

Related papers

Volumes and hyperareas of the spaces of separable and nonseparable qubit-qutrit systems: Initial numerical analyses

Paul B. Slater

Paralleling our recent computationally-intensive work for the case N=4 (quant-ph/0308037), we undertake the task for N=6 of computing to high numerical accuracy, the formulas of Sommers and Zyczkowski (quant-ph/0304041) for the (N^2-1)-dimensional volume and (N^2-2)-dimensional hyperarea of the (separable and nonseparable) N x N density matrices, based on the Bures (minimal monotone) metric. At the same time, we estimate the unknown volumes and hyperareas based on a number of other monotone metrics of interest. Additionally, we estimate -- but with perhaps unavoidably diminished accuracy -- all these volume and hyperarea quantities, when restricted to the ``small'' subset of 6 x 6 density matrices that are separable (classically correlated) in nature. The ratios of separable to separable plus nonseparable volumes, then, yield corresponding estimates of the ``probabilities of separability''. We are particularly interested in exploring the possibility that a number of the various 35-dimensional volumes and 34-dimensional hyperareas, possess exact values -- which we had, in fact, conjectured to be the case for the qubit-qubit systems (N=4), with the ``silver mean'', sqrt{2}-1, appearing to play a fundamental role as regards the separable states.

arXiv:quant-ph/0405114•May 20, 2004•1 citation

Two-qubit separability probabilities and Beta functions

Paul B. Slater

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 $n=4$, 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 $a=1∕2$, $b=\sqrt{3}$ in the real case, and $a=2\sqrt{6}∕5,b=3∕\sqrt{2}$ in the complex case. Assuming the full applicability of these specific incomplete Beta functions, we undertake separable-volume calculations.

arXiv:quant-ph/0609006•Physical Review A•Mar 20, 2007•12 citations

Analytic Fits to Separable Volumes and Probabilities for Qubit-Qubit and Qubit-Qutrit Systems

Paul B. Slater

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.

arXiv:quant-ph/0602109•Feb 14, 2006•0 citations

Metric-dependent probabilities that two qubits are separable

Paul B. Slater

In a previous study (quant-ph/0207181), we formulated a conjecture that arbitrarily coupled qubits (describable by 4 x 4 density matrices) are separable with an a priori probability of 8/(11 π^2) = 0.0736881. For this purpose, we employed the normalized volume element of the Bures (minimal monotone) metric as a probability distribution over the fifteen-dimensional convex set of 4 x 4 density matrices. Here, we provide further/independent (quasi-Monte Carlo numerical integration) evidence of a stronger nature (giving an estimate of 0.0736858 vs. 0.0737012 previously) for this conjecture. Additionally, employing a certain ansatz, we estimate the probabilities of separability based on certain other monotone metrics of interest. However, we find ourselves, at this point, unable to convincingly conjecture exact simple formulas for these new (smaller) probabilities.

arXiv:quant-ph/0306132•Jun 19, 2003•0 citations

Dimension-Independent Positive-Partial-Transpose Probability Ratios

Paul B. Slater

We conduct quasi-Monte Carlo numerical integrations in two very high (80 and 79)-dimensional domains -- the parameter spaces of rank-9 and rank-8 qutrit-qutrit (9 x 9) density matrices. We, then, estimate the ratio of the probability -- in terms of the Hilbert-Schmidt metric -- that a generic rank-9 density matrix has a positive partial transpose (PPT) to the probability that a generic rank-8 density matrix has a PPT (a precondition to separability/nonentanglement). Close examination of the numerical results generated -- despite certain large fluctuations -- indicates that the true ratio may, in fact, be 2. Our earlier investigation (eprint quant-ph/0410238) also yielded estimates close to 2 of the comparable ratios for qubit-qubit and qubit-qutrit pairs (the only two cases where the PPT condition fully implies separability). Therefore, it merits conjecturing (as Zyczkowski was the first to do) that such Hilbert-Schmidt (rank-NM/rank-(NM-1)) PPT probability ratios are 2 for all NM-dimensional quantum systems.

arXiv:quant-ph/0505093•May 13, 2005•1 citation

Qubit-qudit separability/PPT-probability investigations, including Lovas-Andai formula advancements

Paul B. Slater

We begin by advancing conjectures, closely fitting numerical estimates, for the Hilbert-Schmidt separability/PPT probabilities of the 35- and 63-dimensional convex sets of ($2 \times 3$) qubit-qutrit and ($2 \times 4$) qubit-qudit states, as well as certain related scenarios. We reason that the underlying volume formulas might closely parallel those now well-established--via a Lovas-Andai formula--in their lower (15)-dimensional two-qubit analogue. The qubit-qutrit conjecture we obtain is $(\frac{3}{10})^3 =\frac{3^3}{2^3 \cdot 5^3} = \frac{27}{1000} =0.027$, with the counterpart two-qubit separability probablity being $\frac{8}{33}=\frac{2^3}{3 \cdot 11} \approx 0.242424$. In the ($2 \times 4$) qubit-qudit case, we advance $\frac{16}{12375} =\frac{4^2}{3^2 \cdot 5^3 \cdot 11} \approx 0.001292929$ as the PPT-probability. The numerators of the probabilities for $2 \times m$ scenarios are, thus, possibly simply powers of $m$. We also search for the rebit-retrit counterpart to the established two-rebit result of $\frac{29}{64} =\frac{29}{2^6} \approx 0.453125$. Additionally, we find that the Hilbert-Schmidt separability/PPT-probabilities for the two-rebit, rebit-retrit and two-retrit $X$-states all equal $\frac{16}{3 \pi^2} \approx 0.54038$, as well as more generally, that the probabilities based on induced measures are equal across these three sets of $X$-states. Then, we extend the Lovas-Andai two-qubit framework from Hilbert-Schmidt measures to induced ones, raising the possibility of a still more general master formula.

•Mar 28, 2018•0 citations
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 (N2\ensuremath−1)({N}^{2}\ensuremath{-}1)(N2\ensuremath−1)-dimensional volume and (N2\ensuremath−2)({N}^{2}\ensuremath{-}2)(N2\ensuremath−2)-dimensional hyperarea of the (separable and nonseparable) N\ifmmode×\else\texttimes\fiNN\ifmmode\times\else\texttimes\fi{}NN\ifmmode×\else\texttimes\fiN 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.
Paralleling our recent computationally-intensive work for the case N=4 (quant-ph/0308037), we undertake the task for N=6 of computing to high numerical accuracy, the formulas of Sommers and Zyczkowski (quant-ph/0304041) for the (N^2-1)-dimensional volume and (N^2-2)-dimensional hyperarea of the (separable and nonseparable) N x N density matrices, based on the Bures (minimal monotone) metric. At the same time, we estimate the unknown volumes and hyperareas based on a number of other monotone metrics of interest. Additionally, we estimate -- but with perhaps unavoidably diminished accuracy -- all these volume and hyperarea quantities, when restricted to the ``small'' subset of 6 x 6 density matrices that are separable (classically correlated) in nature. The ratios of separable to separable plus nonseparable volumes, then, yield corresponding estimates of the ``probabilities of separability''. We are particularly interested in exploring the possibility that a number of the various 35-dimensional volumes and 34-dimensional hyperareas, possess exact values -- which we had, in fact, conjectured to be the case for the qubit-qubit systems (N=4), with the ``silver mean'', sqrt{2}-1, appearing to play a fundamental role as regards the separable states.
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 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.
In a previous study (quant-ph/0207181), we formulated a conjecture that arbitrarily coupled qubits (describable by 4 x 4 density matrices) are separable with an a priori probability of 8/(11 π^2) = 0.0736881. For this purpose, we employed the normalized volume element of the Bures (minimal monotone) metric as a probability distribution over the fifteen-dimensional convex set of 4 x 4 density matrices. Here, we provide further/independent (quasi-Monte Carlo numerical integration) evidence of a stronger nature (giving an estimate of 0.0736858 vs. 0.0737012 previously) for this conjecture. Additionally, employing a certain ansatz, we estimate the probabilities of separability based on certain other monotone metrics of interest. However, we find ourselves, at this point, unable to convincingly conjecture exact simple formulas for these new (smaller) probabilities.
We conduct quasi-Monte Carlo numerical integrations in two very high (80 and 79)-dimensional domains -- the parameter spaces of rank-9 and rank-8 qutrit-qutrit (9 x 9) density matrices. We, then, estimate the ratio of the probability -- in terms of the Hilbert-Schmidt metric -- that a generic rank-9 density matrix has a positive partial transpose (PPT) to the probability that a generic rank-8 density matrix has a PPT (a precondition to separability/nonentanglement). Close examination of the numerical results generated -- despite certain large fluctuations -- indicates that the true ratio may, in fact, be 2. Our earlier investigation (eprint quant-ph/0410238) also yielded estimates close to 2 of the comparable ratios for qubit-qubit and qubit-qutrit pairs (the only two cases where the PPT condition fully implies separability). Therefore, it merits conjecturing (as Zyczkowski was the first to do) that such Hilbert-Schmidt (rank-NM/rank-(NM-1)) PPT probability ratios are 2 for all NM-dimensional quantum systems.
We begin by advancing conjectures, closely fitting numerical estimates, for the Hilbert-Schmidt separability/PPT probabilities of the 35- and 63-dimensional convex sets of (2×32 \times 32×3) qubit-qutrit and (2×42 \times 42×4) qubit-qudit states, as well as certain related scenarios. We reason that the underlying volume formulas might closely parallel those now well-established--via a Lovas-Andai formula--in their lower (15)-dimensional two-qubit analogue. The qubit-qutrit conjecture we obtain is (310)3=3323⋅53=271000=0.027(\frac{3}{10})^3 =\frac{3^3}{2^3 \cdot 5^3} = \frac{27}{1000} =0.027(103​)3=23⋅5333​=100027​=0.027, with the counterpart two-qubit separability probablity being 833=233⋅11≈0.242424\frac{8}{33}=\frac{2^3}{3 \cdot 11} \approx 0.242424338​=3⋅1123​≈0.242424. In the (2×42 \times 42×4) qubit-qudit case, we advance 1612375=4232⋅53⋅11≈0.001292929\frac{16}{12375} =\frac{4^2}{3^2 \cdot 5^3 \cdot 11} \approx 0.0012929291237516​=32⋅53⋅1142​≈0.001292929 as the PPT-probability. The numerators of the probabilities for 2×m2 \times m2×m scenarios are, thus, possibly simply powers of mmm. We also search for the rebit-retrit counterpart to the established two-rebit result of 2964=2926≈0.453125\frac{29}{64} =\frac{29}{2^6} \approx 0.4531256429​=2629​≈0.453125. Additionally, we find that the Hilbert-Schmidt separability/PPT-probabilities for the two-rebit, rebit-retrit and two-retrit XXX-states all equal 163π2≈0.54038\frac{16}{3 \pi^2} \approx 0.540383π216​≈0.54038, as well as more generally, that the probabilities based on induced measures are equal across these three sets of XXX-states. Then, we extend the Lovas-Andai two-qubit framework from Hilbert-Schmidt measures to induced ones, raising the possibility of a still more general master formula.