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 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.
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 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=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.
Jakobczyk and Siennicki studied two-dimensional sections of a set of (generalized) Bloch vectors corresponding to n x n density matrices of two-qubit systems (that is, the case n = 4). They found essentially five different types of (nontrivial) separability regimes. We compute the Euclidean/Hilbert-Schmidt (HS) separability probabilities assigned to these regimes, and conduct parallel two-dimensional sectional analyses for the higher-level cases n=6,8,9 and 10. Making use of the newly-introduced capability for integration over implicitly defined regions of version 5.1 of Mathematica -- also fruitfully used in our n=4 three-parameter entropy-maximization-based study quant-ph/0507203 -- we obtain a wide-ranging variety of exact HS-probabilities. For n>6, the probabilities are those of having a partial positive transpose (PPT). For the n=6 case, we also obtain biseparability probabilities; in the n=8,9 instances, bi-PPT probabilities; and for n=8, tri-PPT probabilities. By far, the most frequently recorded probability for n>4 is π/4 = 0.785398. We also conduct a number of related analyses, pertaining to the (one-dimensional) boundaries (both exterior and interior) of the separability and PPT domains, and attempt (with limited success) some exact calculations pertaining to the 9-dimensional (real) and 15-dimensional (complex) convex sets of two-qubit density matrices -- for which HS-separability probabilities have been conjectured, but not verified.
An important variable in the 2017 analysis of Lovas and Andai, formally establishing the Hilbert-Schmidt separability probability conjectured by Slater of 6429 for the 9-dimensional convex set of two-rebit density matrices, was the ratio (ε=σ1σ2) of the two singular values (σ1≥σ2≥0) of D221D1−21. There, D1 and D2 were the diagonal 2×2 blocks of a 4×4 two-rebit density matrix ρ. Working within the Lovas-Andai "separability function" (χ~d(ε)) framework, Slater was able to verify further conjectures of Hilbert-Schmidt separability probabilities of 338 and 32326 for the 15-dimensional and 26-dimensional convex sets of two-qubit and two-quater[nionic]-bit density matrices. Here, we investigate the behavior of the three singular value ratios of V=D221D1−21, where now D1 and D2 are the 3×3 diagonal blocks of 6×6 rebit-retrit and qubit-qutrit density matrices randomly generated with respect to Hilbert-Schmidt measure. Further, we initiate a parallel study employing 8×8 density matrices. The motivation for this analysis is the conjectured relevance of these singular values in suitably extending χ~d(ε) to higher dimensional systems--an issue we also approach using certain novel numeric means. Section 3.3 of the 2017 A. Lovas doctoral dissertation (written in Hungarian) appears germane to such an investigation.
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 (2×3) qubit-qutrit and (2×4) 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 (103)3=23⋅5333=100027=0.027, with the counterpart two-qubit separability probablity being 338=3⋅1123≈0.242424. In the (2×4) qubit-qudit case, we advance 1237516=32⋅53⋅1142≈0.001292929 as the PPT-probability. The numerators of the probabilities for 2×m scenarios are, thus, possibly simply powers of m. We also search for the rebit-retrit counterpart to the established two-rebit result of 6429=2629≈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 3π216≈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.
We implement a procedure-based on the Wishart-Laguerre distribution-recently outlined by {Ż}yczkowski and Khvedelidze, Rogojin and Abgaryan, for the generation of random (complex or real) N×N density matrices of rank k≤N with respect to Hilbert-Schmidt (HS) measure. In the complex case, one commences with a Ginibre matrix A of dimensions k×k+2(N−k), while for a real scenario, one employs a Ginibre matrix B of dimensions k×k+1+2(N−k). Then, the k×k product AA† or BBT is diagonalized-padded with zeros to size N×N-and rotated, obtaining a random density matrix. Implementing the procedure for rank-4 rebit-retrit states, for 800 million Ginibre-matrix realizations, 6,192,047 were found separable, for a sample probability of .00774006-suggestive of an exact value 5000387=23⋅5432⋅43=.0774. A conjecture for the HS separability probability of rebit-retrit systems of full rank is 6561860=3822⋅5⋅43≈0.1310775 (the two-rebit counterpart has been proven to be 6429=2629). Subject to these conjectures, the ratio of the rank-4 to rank-6 probabilities would be 100000059049=26⋅56310≈0.059049, with the common factor 43 cancelling. As to the intermediate rank-5 probability, a 2006 theorem of Szarek, Bengtsson and {Ż}ycskowski informs us that it must be one-half the rank-6 probability-itself conjectured to be 100027=23⋅5333, while for rank 3 or less, the associated probabilities must be 0 by a 2009 result of Ruskai and Werner. We are led to re-examine a 2005 qubit-qutrit analysis of ours, in these regards, and now find evidence for a 267370=35⋅112⋅5⋅7≈0.0261878 rank-4 to rank-6 probability ratio.
We seek to derive the probability—expressed in terms of the Hilbert–Schmidt (Euclidean or flat) metric—that a generic (nine-dimensional) real two-qubit system is separable, by implementing the well-known Peres–Horodecki test on the partial transposes (PTs) of the associated 4 × 4 density matrices (ρ). But the full implementation of the test—requiring that the determinant of the PT be nonnegative for separability to hold—appears to be, at least presently, computationally intractable. So, we have previously implemented—using the auxiliary concept of a diagonal-entry-parameterized separability function (DESF)—the weaker implied test of nonnegativity of the six 2 × 2 principal minors of the PT. This yielded an exact upper bound on the separability probability of . Here, we piece together (reflection-symmetric) results obtained by requiring that each of the four 3 × 3 principal minors of the PT, in turn, be nonnegative, giving an improved/reduced upper bound of . Then, we conclude that a still further improved upper bound of can be found by similarly piecing together the (reflection-symmetric) results of enforcing the simultaneous nonnegativity of certain pairs of the four 3 × 3 principal minors. Numerical simulations—as opposed to exact symbolic calculations—indicate, on the other hand, that the true probability is certainly less than . Our analyses lead us to suggest a possible form for the true DESF, yielding a separability probability of , while the absolute separability probability of provides the best exact lower bound established so far. In deriving our improved upper bounds, we rely repeatedly upon the use of certain integrals over cubes that arise. Finally, we apply an independence assumption to a pair of DESFs that comes close to reproducing our numerical estimate of the true separability function.