Numerical and Exact Analyses of Bures and Hilbert-Schmidt Separability and PPT-Probabilities

Numerical and Exact Analyses of Bures and Hilbert-Schmidt Separability and PPT-Probabilities

Paul B. Slater slater@kitp.ucsb.edu Kavli Institute for Theoretical Physics, University of California, Santa Barbara, CA 93106-4030
July 27, 2019
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 . The two-qubit probabilities based, alternatively, on the Hilbert-Schmidt (HS) and operator monotone function measures are “morally” known to be and , respectively. Further, Lovas and Andai have, in fact, proven that the corresponding pair of two-rebit probabilities is 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 HS qubit-qutrit and rebit-retrit separability probabilities are and , 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. The value of is a well-fitting value to an estimate of 0.00139884. Interesting values ( and ) are advanced for the HS and Bures qubit-qudit () PPT-probabilities. We find certain generalized qubit-qutrit scenarios in which the master Lovas-Andai formulas play the same separability function roles as in their original generalized two-qubit settings–obtaining in two distinct 10-dimensional rebit-retrit cases, the same probability of .

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
pacs:
Valid PACS 03.67.Mn, 02.50.Cw, 02.40.Ft, 02.10.Yn, 03.65.-w
preprint:

I Introduction

It has now been formally proven by Lovas and Andai (Lovas and Andai, 2017, Thm. 2) that the separability probability with respect to Hilbert-Schmidt (flat/Euclidean/Frobenius) measure Życzkowski and Sommers (2003) (Bengtsson and Życzkowski, 2017, sec. 13.3) of the 9-dimensional convex set of two-rebit states Caves et al. (2001) is . Additionally, the multifaceted evidence Slater (2018a); Khvedelidze and Rogojin (2018); Milz and Strunz (2014); Fei and Joynt (2016); Shang et al. (2015); Slater (2013); Slater and Dunkl (2012); Slater (2007)–including a recent “master” extension Slater (2018a, b) of the Lovas-Andai framework to generalized two-qubit states–is strongly compelling that the corresponding value for the 15-dimensional convex set of two-qubit states is (with that of the 27-dimensional convex set of two-quater[nionic]bits being [cf. Adler (1995)], among other still higher-dimensional companion random-matrix related results). A still further extension to the use of induced measures–reducing to the Hilbert-Schmidt case for the case –has been found (Slater, 2018b, sec. XII)–yielding, for example, for . (Certainly, one can, however, still aspire to a yet greater “intuitive” understanding of these assertions, particularly in some “geometric/visual” sense [cf. Szarek et al. (2006); Samuel et al. (2018); Avron and Kenneth (2009); Braga et al. (2010); Gamel (2016); Jevtic et al. (2014)], 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-qutrit and 20-dimensional rebit-retrit states are and , respectively (Slater, 2018b, eqs. (15),(20)) (Milz and Strunz, 2014, eq. (33)). (These will be further examined in sec. III 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 Aubrun and Szarek (2017), employing the quite different concepts of 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 Życzkowski, Horodecki, Sanpera and Lewenstein Życzkowski et al. (1998) (cf. Petz and Sudár (1996); Rexiti et al. (2018a); Singh et al. (2014); Batle and Abdel-Aty (2014)). The present author was, then, led–pursuing an interest in “Bayesian quantum mechanics” Slater (1994, 1995) and the concept of a “quantum Jeffreys prior” Kwek et al. (1999)–to investigate how such separability probabilities might depend upon the choice of various possible measures on the quantum states Petz and Sudár (1996).

Of particular initial interest was the the Bures/statistical distinguishability (minimal monotone) measure Slater (2000); Sarkar and Kumar (2019); Šafránek (2017); Forrester and Kieburg (2016); Braunstein and Caves (1994). (“The Bures metric plays a distinguished role since it is the only metric which is also monotone, Fisher- adjusted, Fubini-Study-adjusted, and Riemannian” Forrester and Kieburg (2016). 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” Bej and Deb (2018).)

In (Slater, 2018a, sec. VII.C), we recently reported, building upon analyses of Lovas and Andai (Lovas and Andai, 2017, sec. 4), a two-qubit separability probability equal to . This was based on another (of the infinite family of) operator monotone functions, namely . (The Bures measure itself is associated with the operator monotone function .) (Let us note that the complementary “entanglement probability” is simply . There appears to be no intrinsic reason to prefer/privilege one of these two forms (separability, entanglement) of probability to the other [cf. Dunkl and Slater (2015)]. We observe that the variable denoted , equalling , for , is frequently employed as an upper limit of integration in the Penson-Życzkowski paper, “Product of Ginibre matrices: Fuss-Catalan and Raney distributions” (Penson and Życzkowski, 2011, eqs. (2), (3)).)

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 is a special distribution on the unit ball in operator norm of 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” (Lovas and Andai, 2017, p. 2).

Of central importance to us here will be the construction of Osipov, Sommers and Żyzckowski of an interpolation between the generation of random density matrices with respect to Hilbert-Schmidt and those with respect to Bures measures (Al Osipov et al., 2010, eq. (24)) (cf. (Borot and Nadal, 2012, eq. (33))). This formula takes the form

(1)

where , with yielding a Hilbert-Schmidt density matrix , and , the Bures counterpart . Here, is an complex-valued random matrix pertaining to the Ginibre ensemble (with real and imaginary parts of each of the entries being independent standard normal random variates). Further, is a random unitary matrix distributed according to the Haar measure on . (Of course, in the basic two-qubit case of first interest here.)

It is an intriguing hypothesis that the Bures two-qubit separability probability also assumes a strikingly elegant form (such as , ). (“Observe that the Bures volume of the set of mixed states is equal to the volume of an -dimensional hemisphere of radius (Bengtsson and Życzkowski, 2017, p. 415). It is also noted there that times the area-volume ratio asymptotically increases with the dimensionality , which is typical for hemispheres.)

In the relatively early work Slater (2002), we had conjectured a Bures two-qubit separability probability equal to . But it was later proposed in Slater (2005), in part motivated by the lower-dimensional exact Bures results reported in Slater (2000), that the value might be , where is the “silver mean”. Both of these studies Slater (2002, 2005) were conducted using quasi-Monte Carlo procedures, before the reporting of the indicated Osipov-Sommers-Życzkowski methodology for generating density matrices, random with respect to Bures measure Al Osipov et al. (2010). More recently, in (Slater, 2018b, sec. X.B.1), we reported, using this Ginibre ensemble-based formula (1) an estimate of 0.0733181043 based on 4,372,000,000 realizations, using simply standard (independent) random normal variate generation. (Khvedelidze and Rogojin gave a value of 0.0733 (Khvedelidze and Rogojin, 2018, Table 1) Khv ().)

Performing a parallel (but much smaller) computation in the two-rebit case, based on forty million random density matrices (6,286,209 of them being separable), we obtained a corresponding (slightly corrected) Bures separability probability estimate of 0.1571469. (In doing so, we took, as required, the now real-entried Ginibre matrix A to be (Al Osipov et al., 2010, eqs. (24), (28)), and not as in the two-qubit calculation.)

A phenomenon apparently restricted to the Hilbert-Schmidt case, is that the positive-partial-transpose (PPT) states are equally divided probability-wise between those for which the determinant of the partial transpose of the density matrix () exceeds the determinant of the density matrix itself and vice versa. (Also, along somewhat similar lines, the Hilbert-Schmidt PPT-probability for minimally degenerate [zero eigenvalue] states is half that for nondegenerate states Szarek et al. (2006). The PPT-property is, of course, equivalent to separability for and density matrices (Bengtsson and Życzkowski, 2017, sec. 16.6.C).) Contrastingly, based on some 122,000,000 two-qubit density matrices randomly generated with respect to Bures measure, of the 8,945,951 separable ones found, 5,894,648 of them (that is, 65.89) had , clearly distinct from simply one-half of them (cf. (Khvedelidze and Rogojin, 2018, Tabs. 1, 2) Khv ()).

A formula , for was given in (Slater, 2018c, p. 26). It yields that part of the total separability probability, , for generalized (real [], complex [], quaternionic [],…) two-qubit states endowed with random induced measure, for which the determinantal inequality holds. Here denotes a density matrix, obtained by tracing over the pure states in -dimensions, and , its partial transpose. Further, is a Dyson-index-like parameter with for the standard (15-dimensional) convex set of (complex) two-qubit states.

Further, in the specific Hilbert-Schmidt case (), we can apparently employ (Slater, 2018c, p. 26)

(2)

That is, for , we obtain the previously reported Hilbert-Schmidt formulas, with (the real case) , (the standard complex case) , and (the quaternionic case) —the three simply equalling . More generally, gives that portion, for random induced measure, parameterized by , of the total separability/PPT-probability for which the determinantal inequality holds (Slater, 2018a, eq. (84)). (The [Dyson-index] symbol here is distinct from the use of it below as the length of a sequence in the quasirandom procedure.)

Ii 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 jointly generated by employing low-discrepancy (quasi-Monte Carlo) sequences Leobacher and Pillichshammer (2014). In particular, we have employed an “open-ended” sequence (based on extensions of the golden ratio Livio (2008)) recently introduced by Martin Roberts in the detailed “blog post”, “The Unreasonable Effectiveness of Quasirandom Sequences” Rob (a).

Roberts notes: “The solution to the -dimensional problem, depends on a special constant , where is the value of the smallest, positive real-value of x such that”

(3)

(, yielding the golden ratio, and , the “plastic constant” Rob (b)). The -th terms in the quasirandom (Korobov) sequence take the form

(4)

where we have the -dimensional vector,

(5)

The additive constant is typically taken to be 0. “However, there are some arguments, relating to symmetry, that suggest that is a better choice,” Roberts observes.

These points (5), lying in the -dimensional hypercube , can be converted to (quasirandomly distributed) normal variates, required for implementation of the Osipov-Sommers-Życzkowski formula (1), using the inverse of the cumulative distribution function (Devroye, 1986, Chap. 2). Impressively, in this regard, Henrik Schumacher developed for us a specialized algorithm that accelerated the default Mathematica command InverseCDF for the normal distribution approximately ten-fold, as reported in the highly-discussed post Sch ()–allowing us to vastly increase the realization rate.

We take and 64 in the Roberts methodology, using the Osipov-Sommers-Życzkowski (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 , and the other 32, for the unitary matrix . (A subsidiary question–which appeared in the discussion with Roberts Rob (b)–is the relative effectiveness of employing–to avoid possible “correlation” effects–the same 32-dimensional sequence but at different ’s for and , 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 matrix A, and the other 16 for an orthogonal matrix .

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 density matrices Tilma et al. (2002)), in which 9- and 15-dimensional integration problems were addressed Slater (2005, 2009) (cf. Maziero (2015)). 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 Exp (), 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 Smart et al. (2019), “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 in (4), 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 and (as suggested by the WolframAlpha.com site) . (Informally, Charles Dunkl wrote: ”I would hate to think that the answer is - that is just ugly. One hopes for a rational number.”) At 10,850,000,000 iterations, interestingly, the estimate of 0.0733137814 matched 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 Slater (2018a); Khvedelidze and Rogojin (2018); Milz and Strunz (2014); Fei and Joynt (2016); Shang et al. (2015); Slater (2013); Slater and Dunkl (2012); Slater (2007).

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 . 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 (Lovas and Andai, 2017, Thm. 2).

In Figs. 1 and 2 we show the development of the Bures separability probability estimation procedure in the two cases at hand. (Much earlier versions of these [] plots–together with [less intensive] estimates using –were displayed as Figs. 5 and 6 in Slater (2018b).)

Figure 1: Two-qubit Bures separability probability estimates–divided by –as a function of the number of iterations of the quasirandom procedure, using . 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.
Figure 2: Two-rebit Bures separability probability estimates as a function of the number of iterations of the quasirandom procedure, using . Estimates are recorded at intervals of five million iterations.

Iii Examination of Hilbert-Schmidt Qubit-Qutrit and Rebit-Retrit Separability Conjectures

Based on extensive (standard) random sampling of independent normal variates, in (Slater, 2018b, eqs. (15),(20)), we have conjectured that the Hilbert-Schmidt separability probabilities for the 35-dimensional qubit-qutrit and 20-dimensional rebit-retrit states are (also interestingly rational-valued) and , 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 for this probability (Milz and Strunz, 2014, eq. (33)), while Khvedelidze and Rogojin reported an estimate of 0.0270 (Khvedelidze and Rogojin, 2018, Tab. 1)–but also only 0.0014 for the Bures counterpart [sec. VI].) 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 confidence interval is .

Applying the quasirandom methodology here to further appraise this pair of conjectures, we obtain Figs. 3 and 4. (We take the dimensions of the sequences of normal variates generated to be 72 and 42, respectively.)

Figure 3: Qubit-qutrit Hilbert-Schmidt separability probability estimates–divided by –as a function of the number of iterations of the quasirandom procedure, using . Estimates are recorded at intervals of five million iterations.
Figure 4: Rebit-retrit Hilbert-Schmidt separability probability estimates–divided by –as a function of the number of iterations of the quasirandom procedure, using . Estimates are recorded at intervals of five million iterations.

Interestingly, as in Fig. 1, we observe some drift away–with increasing iterations– from early particularly close fits to the two conjectures. But, as in Fig. 1–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 (Slater, 2018a, App. B), we reported an effort to extend the innovative framework of Lovas and Andai Lovas and Andai (2017) to such qubit-qutrit and rebit-retrit settings. One aspect of interest pertaining to the original density matrix study of Lovas and Andai Lovas and Andai (2017), was that it (surprisingly) appeared possible in Slater (2018a) to extend the original Lovas-Andai framework by restricting our analyses to density matrices in which the two diagonal blocks were themselves diagonal.

Iv An 8-dimensional (-states) rebit-retrit scenario

Along such lines, let us consider an 8-dimensional (-states) rebit-retrit scenario, in which the only non-zero entries are those on the diagonal and anti-diagonal–so that the two diagonal blocks are themselves diagonal. Also, let us employ the ”separability function” framework developed in (Slater, 2008, eq. (5)), where the variable was employed.

Then, with the use of the Mathematica command GenericCylindricalDecomposition, we are able to establish that the associated rebit-retrit Hilbert-Schmidt separability probability is Dunkl and Slater (2015). (This value also holds for the two-rebit and two-retrit -states, while is the two-qubit -states probability (Slater, 2018b, sec. VIII).) The value is obtained by taking the ratio of

(6)

to

(7)

where plays the role of separability function, and is the added factor in the first of the two integrands immediately above.

V Application in higher dimensions of master Lovas-Andai generalized two-qubit formulas

We investigated extending this 8-dimensional rebit-retrit analysis to a 10-dimensional one, by replacing two previously zero entries, so that the two off-diagonal blocks now themselves form X-patterns. The counterpart of the denominator function (7) is, then,

(8)

We, then, need to find the appropriate separability function–corresponding to in (6)–to insert into this integrand–for the numerator–to complete the calculation of the separability probability ratio. In this regard, we were able to preliminarily utilize a sub-optimal separability function (based on the enforecment of the positivity of a submatrix of the partial transpose–but not yet the determinant)

(9)

which yields an upper bound on the separability probability of .

Then, we were able to construct the actual separability function Hei (); Stu (),

(10)

where the dilogarithm is indicated, and . The corresponding separability probability was, then, shown to be Stu ()

(11)

(We have also found very strongly convincing numerical evidence that the same separability probability holds, if instead of considering ten-dimensional rebit-retrit scenarios in which the two off-diagonal blocks have -patterns, one considers that the two diagonal blocks do.)

Further, it is remarkable that the 10-dimensional rebit-retrit separability function (10) turned out to be completely identical with the (polylogarithmic) Lovas-Andai two-rebit function (Slater, 2018a, eq.(2)) (Lovas and Andai, 2017, eq. (9)).

Then, in light of this finding, it is certainly reasonable to entertain an hypothesis that the Lovas-Andai two-qubit function and two-quaterbit function play parallel roles when the associated sets of density matrices share the same zero-nonzero pattern as the two ten-dimensional sets of rebit-retrit density matrices just considered (those with either the two off-diagonal or the two diagonal blocks having -patterns).

Pursuing such an hypothesis, and employing polar and “hyper-polar” coordinates in the very same manner as was done in Slater (2018a), we can readily perform computations leading to a presumptive qubit-qutrit separability probability of and a quaterbit-quatertrit separability probablility of . (Let us interestingly note that , , and . Also, we will note that .)

We have tried directly computing/approximating–an apparently rather challenging task–the hypothesized value of –that is, without simply assuming the applicability of , but have only obtained a value of 0.67696 qub ().

An exploratory (”upgraded”) application Tes () of in the context of the unrestricted twenty-dimensional rebit-retrit setting (rather than the ten-dimensional ones above) led to a separabiity probability calculation of , clearly far different from the 0.1310848 of sec. III. (We note that and .)

Further, we have investigated a twelve-dimensional rebit-retrit scenario, in which all four blocks (rather than just two) now have -patterns. An estimate of 0.5483049 was obtained for the associated separability probability. We have been able to obtain an exact upper bound–by enforcing the positivity of a minor of the partial transpose–of (observing that , , and, of course, ), with the use of the separability function 12D (). (However, despite substantial efforts, we have not yet been able to also enforce the positivity of the determinant of the partial transpose–and, thus, arrive at the governing separability probability itself.)

Vi Bures Qubit-Qutrit Analysis

In Table 1 of their recent study, ”On the generation of random ensembles of qubits and qutrits: Computing separability probabilities for fixed rank states” Khvedelidze and Rogojin (2018), 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 now equal to ), in implementing the Osipov-Sommers-Życzkowski formula (1) given above with . (For the companion Bures rebit-retrit estimation, we would have a smaller , that is, 78–but, given our Bures two-rebit analysis above, we were not optimistic in being able to advance a possible exact value.) In Fig. 5 we show a (scaled) 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 3,174 million iterations) was . This can be well-fitted by .

Figure 5: Qubit-qutrit Bures separability probability estimates–divided by –as a function of the number of iterations of the quasirandom procedure, using . Estimates are recorded at intervals of one million iterations.

It would now seem of interest to examine a second 10D rebit-retrit scenario, in which rather than the two off-diagonal blocks having -state form, the two diagonal blocks would.

vi.1 Higher-Dimensional Bures Analyses

To estimate the Bures qubit-qudit bipartite () PPT-probability, we employed a 256-dimensional quasirandom sequence, obtaining 4,760 PPT density matrices in 830 million realizations, yielding an estimated probability of . An interesting candidate for a possible corresponding exact value is (Fig. 6).

Figure 6: Qubit-qudit () Bures PPT-probability estimates–divided by –as a function of the number of iterations of the quasirandom procedure, using . Estimates are recorded at intervals of one million iterations.

For the Bures two-qutrit scenario, employing a 324-dimensional sequence, only 43 PPT density matrices were generated in 678 million realizations, yielding an estimate of (Fig. 7). (It would be of interest to relate this last very small PPT-probability estimation to the asymptotic analyses of Aubrun and Szarek Aubrun and Szarek (2017), as well as Ye Ye (2009).)

Figure 7: Two-qutrit Bures PPT-probability estimates as a function of the number of iterations of the quasirandom procedure, using . Estimates are recorded at intervals of one million iterations.

Vii Higher-Dimensional Hilbert-Schmidt Analyses

Further, in (Slater, 2018b, sec. 3,5), we had suggested Hilbert-Schmidt PPT-probability hypotheses for the and qubit-qudit systems of and , and and for their rebit-redit analogues. In regard to Hilbert-Schmidt two-qutrit probabilities, an estimate of 0.00010218 based on 100 million random realizations was reported in sec. III.A of “Invariance of Bipartite Separability and PPT-Probabilities over Casimir Invariants of Reduced States” Slater (2016). (An intriguing possible corresponding exact value is –or .) These last results could be re-examined with the use of the quasirandom procedure introduced by Martin Roberts Rob (a).

In fact, for the Hilbert-Schmidt qubit-qudit and two-qutrit scenarios, in ongoing such analyses, we have obtained current estimates of 0.0012928963 and 0.00010275452 based on 2,104 and 1,768 million iterations, respectively (Figs. 8 and 9).

Figure 8: Qubit-qudit () Hilbert-Schmidt PPT-probability estimates–divided by –as a function of the number of iterations of the quasirandom procedure, using . Estimates are recorded at intervals of one million iterations.
Figure 9: Two-qutrit Hilbert-Schmidt PPT-probability estimates as a function of the number of iterations of the quasirandom procedure, using . Estimates are recorded at intervals of one million iterations.

Viii Concluding Remarks

We should stress that the problem of formally deriving the Bures two-rebit and two-qubit separability probabilities, and, thus, testing the candidate value () advanced here, certainly currently seems intractable–even, it would seem, in the pioneering framework of Lovas and Andai Lovas and Andai (2017). (Perhaps some formal advances can be made, in such regards, with respect to -states [cf. Xiong et al. (2017)].)

Let us note that the “master Lovas-Andai” formula for generalized two-qubit Hilbert-Schmidt separability probabilities reported in (Slater, 2018a, sec. VIII.A)

(12)

( being a singular-value ratio, and not the quasirandom dimension parameter–the random-matrix Dyson index) has been recently extended to apply to the still more general class of “induced measures”, giving expressions for Slater (2018b). (Also, we have sought to develop an alternative framework to that of Lovas and Andai, in the context of “Slater separability functions”, but not yet fully successfully Lov (a, b).)

As specific illustrations here–with the assistance of C. Dunkl–are the formulas (Slater, 2018b, sec. B.3.c)–with –for and :

It may be of interest to the reader to here include a response of Martin Roberts to a query as to whether to calculate a 64D integral, it is optimal or not to use a 64D low-discrepancy sequence, as employed above. Roberts interestingly replied: “It depends. In theory, the convergence rate of simple random sampling is O(1/n), whereas for low discrepancy sequences it is O(). The term implies that in theory for some large D, and very large N, the convergence rate of quasirandom sequences is inferior to simple random sampling. However, the classic Big O notation ignores two things, which in practice are crucial. (i) Big O notation is for . For finite N, the constants of proportionality play a big role in determining which one is more efficient. (ii) it has been found that for many high dimensional integrals (especially the finance, computer vision, and natural language processing) although they may outwardly look like high dimensional functions they are in fact really relatively low dimensional problems embedded in a high dimensional manifold. Therefore the pragmatic D in the above expression, is really the ’intrinsic’ D. This is why finance options-pricing which is based on integrations over a few hundred dimensions are still more efficient with quasirandom sampling.”

In section 4 of their recent study Lovas and Andai (2017), Lovas and Andai extended their analyses from one involving the (non-monotone Ozawa (2000)) Hilbert-Schmidt measure to one based on the operator monotone function . They were able to conclude (for the case [a Dyson-type random-matrix index]) that the applicable “separability function” in this case, , is precisely the same as the Hilbert-Schmidt counterpart .

Now, quite strikingly, we obtained Slater (2018a), using this function, for the two-qubit () analysis, the ratio of to , that is,

(13)

(We observe that such results–as with the Hilbert-Schmidt value of –interestingly appear to reach their most simple/elegant in the [standard, 15-dimensional] two-qubit setting, where the off-diagonal entries of the density matrix are, in general, complex-valued.)

Lovas and Andai have shown that the two-rebit separability probability based on the operator monotone function is approximately 0.26223, asserting that the integrand can be evaluate[d] only numerically”. Nevertheless, we investigated–so far, rather not too productively, as with the Bures two-rebit estimate 0.157096234 here–the possibility of finding an exact, underlying value for this statistic. (Our investigation, in this regard, is reported in Sta (). It involved first performing a series expansion of the elliptic and hypergeometric functions in their integrand. We were able to then integrate this series expansion, but only over a restricted range–rather than –of the two indices. Numerical summation over this restricted set yielded a value of only 0.0042727 [reported in Sta ()] vs. 0.26223.)

So, in brief summary, let us state that at this stage of our continuing investigations, it appears that we have a set of three exact-valued measure-dependent two-qubit separability probabilities ( [Hilbert-Schmidt], [operator monotone ], [Bures–minimal monotone ]), but only one two-rebit one ( [Hilbert-Schmidt]).

The [apparent lesser] separability probabilities for other members–Kubo-Mori, Wigner-Yanase,…–of the monotone family have been estimated in Slater (2005)–cf. Singh et al. (2014); Batle and Abdel-Aty (2014). But since there is no present mechanism apparent for generating density matrices random with respect to such other measures [cf. (Puchała and Miszczak, 2011, sec. V.B) in regard to superfidelity], the quasirandom procedure seems unavailable for them. Also the use of measures that are non-monotone in nature–in addition to the well-studied Hilbert-Schmidt one–would be of interest, for example, the Monge Zyczkowski and Slomczynski (2001), the Husimi Slater (2006); Rexiti et al. (2018b) measures. However, separability/PPT-probabilities could be so analyzed for the class of induced measures Życzkowski and Sommers (2001).

Acknowledgements.
This research was supported by the National Science Foundation under Grant No. NSF PHY-1748958. I thank Nicolas Tessore for certain computational contributions.

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