Master Lovas-Andai and Equivalent Formulas Verifying the Two-Qubit Hilbert-Schmidt Separability Probability and Companion Rational-Valued Conjectures
We begin by investigating relationships between two forms of Hilbert-Schmidt two-rebit and two-qubit “separability functions”–those recently advanced by Lovas and Andai (J. Phys. A 50  295303), and those earlier presented by Slater (J. Phys. A 40  14279). In the Lovas-Andai framework, the independent variable is the ratio of the singular values of the matrix formed from the two diagonal blocks () of a density matrix . In the Slater setting, the independent variable is the diagonal-entry ratio –with, of central importance, or when both and are themselves diagonal. Lovas and Andai established that their two-rebit “separability function” () yields the previously conjectured Hilbert-Schmidt separability probability of . We are able, in the Slater framework (using cylindrical algebraic decompositions [CAD] to enforce positivity constraints), to reproduce this result. Further, we newly find its two-qubit, two-quater[nionic]-bit and “two-octo[nionic]-bit” counterparts, , and . These immediately lead to predictions of Hilbert-Schmidt separability/PPT-probabilities of , and , in full agreement with those of the “concise formula” (J. Phys. A 46  445302), and, additionally, of a “specialized induced measure” formula. Then, we find a Lovas-Andai “master formula”, , encompassing both even and odd values of . Remarkably, we are able to obtain the formulas, , applicable to full (9-, 15-, 27-) dimensional sets of density matrices, by analyzing (6-, 9, 15-) dimensional sets, with not only diagonal and , but also an additional pair of nullified entries. Nullification of a further pair still, leads to X-matrices, for which a distinctly different, simple Dyson-index phenomenon is noted. C. Koutschan, then, using his HolonomicFunctions program, develops an order-4 recurrence satisfied by the predictions of the several formulas, establishing their equivalence. A two-qubit separability probability of is obtained based on the operator monotone function , with the use of .
pacs:Valid PACS 03.67.Mn, 02.50.Cw, 02.40.Ft, 02.10.Yn, 03.65.-w
- I Introduction and initial analyses
- II Expanded analyses
- III Relations between and Bloore/Slater variable
IV Scenarios for which or
- IV.1 Seven-dimensional convex set of two-rebit states
- IV.2 Eleven-dimensional convex set of two-qubit states
- IV.3 Common features of two-rebit, two-qubit and two-quaterbit constraint sets
- V X-states functions conform to Dyson-index ansatz
- VI Formula for the Lovas-Andai two-quaterbit function
- VII General Construction of the Lovas-Andai Formulas
- VIII Concluding Remarks
- A Absolute separability probabilities
- B Rebit-retrit and qubit-qutrit analyses
- C Comparison of Ginibre and Cholesky methods for quaternion positive-definite matrices–by C. F. Dunkl
I Introduction and initial analyses
To begin our investigations, focusing on interesting recent work of Lovas and Andai Lovas and Andai (2017), we examined a certain possibility–motivated by a number of previous studies (e.g. Slater (2007a, 2008, 2009a, 2009b)) and the apparent strong relevance there of the Dyson-index vantage upon random matrix theory Dumitriu and Edelman (2002). More specifically, we ask whether the sought Lovas-Andai “separability function” for the standard (complex) two-qubit systems might be simply proportional (or even equal) to the square of their successfully constructed two-rebit separability function (Lovas and Andai, 2017, eq. (9)),
Let us note that has a closed form,
where the polylogarithmic function is defined by the infinite sum
for arbitrary complex and for all complex arguments with . Let us further observe that in the proof of (1), the authors were able to formulate the problem of finding rather concisely in terms of a “defect function” (Lovas and Andai, 2017, App. A),
We will be able in sec. IV.1.1 to obtain the formula (2) for by alternative (cylindrical algebraic decomposition Strzeboński (2016)) means. Further, in sec. IV.2.1, we will apply the same basic methodology to obtain (the much simpler) polynomial formula (42) for . Then, we will be able (sec. VII) to develop a general procedure for finding for integer . The (rational) separability/PPT-probabilities predicted using these functions will–as extensive symbolic and numerical testing reveals–be identically the same as those yielded by the “concise” formula reported in (Slater, 2013, eqs. (1)-(3)),
(Here, . This set of relationships was developed with the [high-precision] use of a probability distribution reconstruction method Provost (2005), using formulas for the moments of the determinants of density matrices and of their partial transposes. This was followed by an application by Qing-Hu Hou of “Zeilberger’s algorithm” (“creative telescoping”) Paule and Schorn (1995) to the large hypergeometric-based expression so-obtained, displayed in Fig. 3 of Slater (2013). See also (84), for a quite distinct, but–as will eventually be shown here–equivalent formula [Fig 24].)
As part of their analysis, Lovas and Andai assert (Lovas and Andai, 2017, p. 13) that
with the denominator evaluating to . Here, is the Hilbert-Schmidt separability probability for the nine-dimensional convex set of two-rebit states Caves et al. (2001). With the indicated use of this probability evaluates to (the numerator of (7) equalling , with ), a result that had been strongly anticipated by prior analyses Fei and Joynt (2016); Slater (2013); Slater and Dunkl (2012).
If the (Dyson-index) proportionality relationship
held, we would have
Here, is–in the Lovas-Andai framework–the Hilbert-Schmidt separability probability for the fifteen-dimensional convex set of the (standard/complex) two-qubit states Gamel (2016). They expressed hope that they too would be able to demonstrate that , as also has been strongly indicated is, in fact, the case Fei and Joynt (2016); Slater (2013); Slater and Dunkl (2012); Shang et al. (2015); Zhou et al. (2012). (”It is interesting whether this observation has some deep background or is an accidental fact only” Khvedelidze and Rogojin (2015).) We generalized (from ) the denominator of the ratio (9) to
Our Dyson-index-based ansatz, then, is that
gives the generalized (-th) Hilbert-Schmidt separability probability. For , we recover the two-rebit formula (7), while for , under the ansatz, we would conjecturally obtain the two-qubit value of , while for , the two-quater[nionic]bit value of would be gotten, and similarly, for , the (presumably) two-octo[nionic]bit value of Slater (2016a). (The volume forms listed in (Lovas and Andai, 2017, Table 1) for the sets of self-adjoint matrices , , are in the case, and in the case, respectively. Our calculations of the term appearing in the several Lovas-Andai volume formulas (Lovas and Andai, 2017, pp. 10, 12), such as this one for the volume of separable states,
[the function being the ratio of the two singular values of the matrix ] appear to be consistent with the use of the terms in the ansatz (11).)
The values themselves correspond to the real, complex, quaternionic and octonionic division algebras. We can, further, look at the other nonnegative (non-division algebra) integral values of . So, for , we have the formal prediction Slater and Dunkl (2012, 2015) of .
In this context, let us first note that for the denominator of (9), corresponding to , we obtain (a result we later importantly employ (43)). Using high-precision numerical integration (https://mathematica.stackexchange.com/q/133556/29989) for the corresponding numerator of (9), we obtained 0.0358226206958479506059638010848. The resultant ratio (dividing by ) is 0.220393076546720789860910104330, within of 0.242424. However, somewhat disappointingly, it was not readily apparent as to what exact values these figures might correspond.
The analogous numerator-denominator ratio in the (two-quaterbit) instance was 0.0534499, while the predicted separablity probability is . It can then be seen that the required constant of proportionality () in the case is not particularly close to the square of that in the instance (). Similarly, in the case, the numerator-denominator ratio is 0.00319505, while the predicted value would be (with the ratio of these two values being 0.293873). So, our ansatz (11) would not seem to extend to the sequence of constants of proportionality themselves conforming to the Dyson-index pattern. But the analyses so far could only address this specific issue concerning constants of proportionality.
Ii Expanded analyses
We, then, broadened the scope of the inquiry with the use of this particular formula of Lovas and Andai for the Hilbert-Schmidt volume of separable states (Lovas and Andai, 2017, p. 11),
Here denotes the upper diagonal block, and , the lower diagonal block of the density matrix (Lovas and Andai, 2017, p. 3),
The Lovas-Andai parameter is defined as 1 in the two-rebit case and 2 in the standard two-qubit case (that is, in our notation, ). Further, the relevant division algebra is , or , according to . The exponential term in (13) corresponds to the “singular value ratio”,
of the matrix , where the Hilbert-Schmidt norm is indicated. (In (Yin et al., 2015, sec. IV) the ratio of singular values of “empirical polarization matrices” is investigated.)
ii.1 Generation of random density matrices
ii.1.1 Two-rebit case
Firstly, taking , we generated 687 million random (with respect to Hilbert-Schmidt measure) density matrices situated in the 9-dimensional convex set of two-rebit states (Życzkowski et al., 2011, App. B) Caves et al. (2001); Batle et al. (2003). Of these, 311,313,185 were separable (giving a sample probability of 0.453149, close to the value of , now formally established by Lovas and Andai). Additionally, we binned the two sets (separable and all) of density matrices into 200 subintervals of , based on their corresponding values of (Fig. 1). Fig. 2 is a plot of the estimated separability probabilities (remarkably close to linear with slope 1–as previously observed (Lovas and Andai, 2017, Fig. 1)), while Fig. 3 shows the result of subtracting from this curve the very well-fitting (as we, of course, expected from the Lovas-Andai proof) function , as given by ((1),(2)). (If one replaces by simply its close approximant , then the corresponding integrations would yield a “separability probability”, not of , but of . If we similarly employ in the two-qubit case, rather than the [previously undetermined] , the corresponding integrations yield , and not the presumed correct result of .) Fig. 21 will serve as the two-qubit analogue of Fig. 3, further validating the formula (42) for to be obtained.
ii.1.2 Two-qubit case
We, next, to test a Dyson-index ansatz, taking , generated 6,680 million random (with respect to Hilbert-Schmidt measure) density matrices situated in the 15-dimensional convex set of (standard) two-qubit states (Życzkowski et al., 2011, eq. (15)). Of these, 1,619,325,156 were separable (giving a sample probability of 0.242414, close to the conjectured, well-supported [but not yet formally proven] value of ) (cf. Singh et al. (2014)). We, again, binned the two sets (separable and all) of density matrices into 200 subintervals of , based on their corresponding values of (Fig. 4). Fig. 5 is a plot (now, clearly non-linear [cf. Fig. 2]) of the estimated separability probabilities, along with the quite closely fitting, but mainly slightly subordinate curve. Fig. 6 shows the result/residuals (of relatively small magnitude) of subtracting from the estimated separability probability curve.
So, it would seem that the square of the explicitly-constructed Lovas-Andai two-rebit separability function provides, at least, an interesting approximation to the sought two-qubit separability function .
The Dyson-index ansatz–the focus earlier in the paper–appears to hold in some trivial/degenerate sense if we employ rather than the Lovas-Andai or Slater separability functions discussed above, the “Milz-Strunz” ones Milz and Strunz (2015). Then, rather than the singular-value ratio or the ratio of diagonal entries , one would use as the dependent/predictor variable, the Casimir invariants of the reduced systems Slater (2016b). In these cases, the separability functions become simply constant in nature. In the two-rebit and two-qubit cases, this invariant is the Bloch radius () of one of the two reduced systems. From the arguments of Lovas and Andai (Lovas and Andai, 2017, Cor. 2, Thm. 2), it appears that one can assert that the Milz-Strunz form of two-rebit separability function assumes the constant value for . Then, it would seem that the two-qubit counterpart would be the constant value for , with the corresponding (Dyson-index ansatz) constant of proportionality being .
Iii Relations between and Bloore/Slater variable
Let us now note a quite interesting phenomenon, apparently relating the Lovas-Andai analyses to previous ones of Slater Slater (2007a). If we perform the indicated integration in the denominator of (7), following the integration-by-parts scheme adopted by Lovas and Andai (Lovas and Andai, 2017, p. 12), at an intermediate stage we arrive at the univariate integrand,
(Its integral over equals the noted value of , where .) This, interestingly, bears a very close (almost identical) structural resemblance to the jacobian/volume-element
(integrating to over ) reported by Slater in (Slater, 2007a, eq. (15)) and (Slater, 2007b, eq. (10)), also in the context of two-rebit separability functions. (We change the notation in those references from to here, since we have made the transformation , to facilitate this comparison, and the analogous one below in the two-qubit context–with the approach of Lovas and Andai. However, we will still note some results below in the original  framework.) To faciltate the comparison between these two functions, we set , and then divide (15) by (16), obtaining the simple ratio
But we note that in in Slater (2007a) and Slater (2007b)–motivated by work in a density matrix context of Bloore Bloore (1976)–the variable was taken to be the ratio of the square root of the product of the (1,1) and (4,4) diagonal entries of the density matrix (Slater, 2007a, eq. (1))
to the product of the (2,2) and (3,3) ones, while in Lovas and Andai (2017), it would be the ratio of the singular values of the noted matrix . From (Slater, 2007a, eq. (91)), we can deduce that one must multiply by , so that its integral from 0 to 1 equals the Lovas-Andai counterpart result of . (The jacobian of the transformation to the two-rebit density matrix parameterization (18) is , and for the two-qubit counterpart, (Slater, 2007b, p. 4). These jacobians are also reported in Andai (2006).)
In (Slater, 2007a, eq. (93)), the two-rebit separability function was taken to be proportional to the incomplete beta function –an apparently much simpler function than the Lovas-Andai counterpart (2) above. Given the just indicated scaling by , to achieve the separability probability numerator result of Lovas and Andai, we must take the hypothesized separability function to be, then, .
A parallel phenomenon is observed in the two-qubit case, where (Slater, 2007b, eq. (11)),
To achieve the Lovas-Andai two-qubit denominator result, we must multiply (19) by 328007680.
The two-qubit separability function advanced in Slater (2007a) was proportional to the square of that––employed in the two-rebit context. Now, to obtain the two-qubit numerator result of necessary for the separability probability outcome, we took the associated separability function to simply be . We refer the reader to Figure 2 in Slater (2007a) (and Figs. 13 and 14 below) to see the extraordinarily good fit of this function. (However, the two-rebit fit displayed there does not appear quite as good.)
Let us now supplement the earlier plots in Slater (2007a), with some newly generated ones. (Those 2007 plots were based on quasi-Monte Carlo [“low-discrepancy” point Bratley et al. (1992)] sampling, while the ones presented here are based on more “state-of-the-art” sampling methods Życzkowski et al. (2011), with many more density matrices [but, of “higher-discrepancy”] generated.) In Figs. 7 and 8 we show the two-rebit separability probabilities as a function, firstly, of and, secondly, as a function of , together with the curves and , respectively.
In Figs. 11, 12, 13 and 14, rather than showing the estimated separability probabilities together with the separability functions as in the previous four figures, we show the estimated separability probabilities minus the separability functions, that is, the residuals from this fits.
So, at this stage, the evidence is certainly strong that the Dyson-index ansatz is at least of some value in approximately fitting the relationships between two-rebit and two-qubit Hilbert-Schmidt separability functions.
iii.1 Formulas linking the Lovas-Andai variable and the Slater/Bloore variable
and inverting it, we find
For the two-qubit counterpart, we have
The ’s are as in the two-rebit case (26), and the ’s are now the corresponding imaginary parts in the natural extension of the two-rebit density matrix parameterization (18). A similar inversion yields
It appears to be a challenging problem, using these relations (13), (22) and (24), to transform the -parameterized volume forms and separability functions in the Lovas-Andai framework to the -parameterized ones in the Slater setting, and vice versa. (The presence of the and variables in the formulas, undermining any immediate one-to-one relationship between and , is a complicating factor.)
The correlation between the and variables, estimated on the basis of one million randomly-generated (with respect to Hilbert-Schmidt measure) density matrices was 0.631937 in the two-rebit instance, and 0.496949 in the two-qubit one.
Also, in these two sets of one million cases, was always larger than . This dominance effect (awaiting formal verification) is reflected in Figs. 15 and 16, being plots of the separability probabilities (again based on samples of size 5,077 and 3,715 million, respectively) as joint functions of and , with no results appearing in the regions .
It has been noted (https://mathoverflow.net/q/262943/47134 that for a diagonal density matrix that (inverting ratios, if necessary, so that both are less than or greater than 1). This equality can also be observed by setting (and ) in the equations immediate above.
Let us now display three plots that support, but only approximately, the possible relevance of the Dyson-index ansatz for two-rebit and two-qubit separability functions. In Fig. 17, we show the ratio of the square of the two-rebit separability probabilities to the two-qubit separability probabilities, in terms of the variable employed by Slater, . In Fig. 18, we show the Lovas-Andai counterpart, that is, in terms of the ratio of singular values variable, . Further, in Fig. 19 we display the ratio of the square of the two-dimensional two-rebit plot (Fig. 15) to the two-dimensional two-qubit plot (Fig. 16). These three figures all manifest an upward trend in the ratios as and/or increase.
Iv Scenarios for which or
iv.1 Seven-dimensional convex set of two-rebit states
If we set in the relation (21), we obtain or . So, let us try to obtain the separability function when these null conditions are fulfilled. First, we found that the Hilbert-Schmidt volume of the seven-dimensional convex set is equal to , with a jacobian for the transformation to equal to
( is the normalization constant corresponding to (Lovas and Andai, 2017, Table 2), appearing in the “defect function” (3), as well as the volume of the standard unit ball in the normed vector space of matrices with real entries, denoted by .)
We were, further, able to impose the condition that two of the principal minors of the partial transpose are positive. The resultant separability function (Fig. 20) was
with an associated separability probability of .
We, then, sought to impose–as both necessary and sufficient for separability Peres (1996); Horodecki et al. (1996)–the positivity of the partial transpose of the density matrix. First, we found that the associated separability function assumes the value 1 at . For , we formulated four-dimensional constrained integration problems. Mathematica reduced them to two-dimensional integration problems, for which we were able to perform high precision calculations. Remarkably, the values obtained agreed (using (2)) with those for and to more than twenty decimal places. The two-dimensional integrands Mathematica yielded for were of the form
So, in light of this evidence, we are confident in concluding that the Lovas-Andai two-rebit separability function serves as both the Lovas-Andai and Slater separability functions in this seven-dimensional setting.
iv.1.1 Reproduction of Lovas-Andai two-rebit separability function
Making use of these last set of relations, we were able to reproduce the Lovas-Andai two-rebit separability function , given in (2). We accomplished this by, first, reducing the (general for integer ) two-dimensional integrands (29) and (30) to two piecewise one-dimensional ones of the form
To obtain these one-dimensional integrands, which we then were able to explicitly evaluate, we made the substitution , then integrated over , with , so that . Let us note that in this approach, the dependent variable () appears in the integrands, while in the Lovas-Andai derivation, the dependent variable () appears as a limit of integration. The counterpart set of two piecewise integrands to (31) and (32) for the reciprocal case of are
over with and
over with . The corresponding univariate integrations then directly yield the Lovas-Andai two-rebit separability function , given in (2), now with , rather than .
As an interesting aside, let us note that we can obtain in (21), in a nontrivial fashion (that is, not just by taking ), by setting
leading to an eight-dimensional framework. However, this result did not seem readily amenable to further study/analysis.
iv.2 Eleven-dimensional convex set of two-qubit states
Let us repeat for the 15-dimensional convex set of two-qubit states, the successful form of analysis in the preceding section, again nullifying the (1,2), (2,1), (3,4), (4,3) entries of , so that the two diagonal blocks are themselves diagonal. This leaves us in an 11-dimensional setting. The associated volume we computed as . (Here is the normalization constant corresponding to (Lovas and Andai, 2017, Table 2), as well as the volume of the standard unit ball in the normed vector space of matrices with complex entries, denoted by .) The associated jacobian for the transformation to the variable is
The imposition of positivity for one of the principal minors of the partial transpose yielded a separability function of for , with an associated bound on the true separability probability of this set of eleven-dimensional two-qubit density matrices of . (This function bears an interesting resemblance to the later reported important one (41). The insertion into (11) of it, in its form, , leads to a separability probability prediction of .)
Again–as in the immediately preceding seven-dimensional two-rebit setting–imposing, as both necessary and sufficient for separability Peres (1996); Horodecki et al. (1996), the positivity of the partial transpose of the density matrix, we find that the associated separability function assumes the value 1 at . For , our best estimate was 0.36848, which in line with the seven-dimensional analysis, would appear to be an approximation to the previously unknown value of