Abstract

Previously, a formula, incorporating a hypergeometric function, for the Hilbert-Schmidt-averaged determinantal moments of density-matrices () and their partial transposes (), was applied with to the generalized two-qubit separability probability question. The formula can, furthermore, be viewed, as we note here, as an averaging over “induced measures in the space of mixed quantum states.” The associated induced-measure separability probabilities () are found—via a high-precision density approximation procedure—to assume interesting, relatively simple rational values in the two-re[al]bit (), (standard) two-qubit (), and two-quater[nionic]bit () cases. We deduce rather simple companion (rebit, qubit, quaterbit, …) formulas that successfully reproduce the rational values assumed for general  . These formulas are observed to share certain features, possibly allowing them to be incorporated into a single master formula.

1. Introduction

The question of the probability that a generic quantum system is separable/disentangled was raised in a 1998 paper of Życzkowski et al. entitled “Volume of the set of separable states” [1]. Certainly, any particular answer to this question will crucially depend upon the measure that is attached to the systems in question. A large body of literature has arisen from the 1998 study, and we seek to make a significant contribution to it, addressing heretofore unsolved problems. Let us point out the work of Aubrun et al. [2], which addresses questions of a somewhat similar nature to those examined below while employing the same class of measures. However, their work is set in an asymptotic framework, while we will be concerned with obtaining exact finite-dimensional results (cf. [3]). On the other hand, Singh et al. [4] did focus on finite-dimensional scenarios but with a distinct form of measure, the one originally used in [1].

We have investigated the possibility of extending to the class of “induced measures in the space of mixed quantum states” [5, 6] the line of analysis reported in [7, 8], the principal separability probability findings of which most notably the two-qubit conjecture of has recently been robustly supported, with the use of extensive Monte-Carlo sampling by Fei and Joynt [9] as well as by Milz and Strunz to somewhat similar effect [10, Figure 4, equations (30), (31)] (cf. [11, Table 1]). This earlier line of work pertained to the use of the Hilbert-Schmidt measure (the particular symmetric case, , of the induced measures) on the high-dimensional convex sets of generalized (real-, complex-, quaternionic-entried) two-qubit () states.

In [7, page 30], a central role had been played by the (not yet formally proven) determinantal moment formula obtained thereon the basis of extensive computations. Here denotes the partial transpose [12] of the density matrix and , its determinant and generalized hypergeometric function notation, is employed. The brackets represent averaging with respect to Hilbert-Schmidt measure [13]. Furthermore, is a random-matrix Dyson-index-like parameter [14], assuming, in particular, the value 1 for the standard (fifteen-dimensional convex set of) density matrices with complex-valued off-diagonal entries.

It subsequently occurred to us that this hypergeometric-based moment formula could be readily adapted to the broader class of random induced measures by considering, in the notation of [5, 6], thatwhere is the dimension of the ancilla/environment state, over which the tracing operation is performed.

As in the earlier work [7, 8], a high-precision density-approximation (inverse) procedure of Provost, incorporating the first 11,401 such determinantal moments, strongly indicates that the random induced-measure separability probabilities () assume interesting, relatively simple rational values in the two-re[al]bit (), (standard) two-qubit (), and two-quater[nionic]bit () cases, particularly so for (Section 2). One striking example is that for ; the separability probability is found to be (to fifteen decimal places). In fact, based on extensive calculations () of this nature, we are able to deduce rather simple companion (rebit, qubit, quaterbit) formulas (3)–(5) that successfully reproduce the rational values assumed for general integer and half-integer (Section 3).

Further efforts along these lines have been given in a subsequent paper [15], in which the determinantal inequality is now imposed rather than the broader inequality . Of course, , while is both a necessary and sufficient condition for separability here [12, 16]. There, equivalent hypergeometric- and difference-equation-based formulas, , for , were given for that (rational-valued) portion of the total separability probability satisfying the stricter inequality. We also preliminarily investigate this problem below in Section 4.

Milz and Strunz [10] have recently reported a highly interesting finding that the conjectured Hilbert-Schmidt separability probability of holds constant along the radius of the Bloch sphere of either of the reduced subsystems of generic two-qubit () systems. We are presently investigating the nature that the separability probability is taken as a joint function of the radii of the two single-qubit subsystems and related questions.

2. Analysis

We pursue the indicated extension of our earlier (Hilbert-Schmidt-based) work to random induced measures, in general. As in [7, 8], the determinantal moment formula above is employed in the Legendre polynomial-based (Mathematica-implemented) density approximation (inverse) procedure of Provost [17]. This possesses a least-squares rationale. The program as originally presented is speeded by incorporating the well-known recursion formula for Legendre polynomials so that successive polynomials do not have to be computed ab initio. The computations are all exact, in nature, rather than numerical. Provost advises that the procedure should be regarded as an “approximation” rather than an “estimation” scheme [17]. Let us note that the implementation of the procedure requires considerable caution and an adaptive strategy when the term [7, Section ] in the underlying summation formula for the hypergeometric-based determinantal moments is zero. It is zero if , that is, if values for which and occur in the summation .

Now, with the use of an unprecedentedly large number (11,401) of the determinantal moments, we found, (to ten decimal places) for , the separability probability of the standard, complex () 15-dimensional convex set of two-qubit states to be . On the other hand, for the Hilbert-Schmidt case (), a very compelling body of evidence of a number of types (though yet no formal proof) has been adduced that the corresponding separability probability, as has been already noted, is [7–10].

For the quaternionic () case, the induced-measure () separability probability (now to thirteen decimal places) was , while the Hilbert-Schmidt counterpart strongly appears to be [7–9].

Let us further note, though any immediate quantum-mechanical random-matrix division-algebra interpretation does not seem at hand for , that for , we obtain a “separability-probability” approximant, based on the 11,401 moments, that, to a remarkable sixteen decimal places equaled . This particularly high accuracy appears to essentially be an artifact of the Legendre polynomial-based procedure that commences with a uniform distribution over the interval . For such a distribution, the probability over the “separability” interval of is the ratio of to , that is, , quite near to 0.0657655. So as separability probability approximants increasingly deviate from the uniform-based one of , at least for specific , we can expect convergence of the density-approximation procedure to relatively weaken.

For the two-rebit scenario (), the associated Hilbert-Schmidt separability probability strongly appears to be [7, 8], while in the random induced-measure counterpart, we obtain (to almost nine decimal places) a value once again larger than that for the Hilbert-Schmidt case of , that is, . Note the powers of 2 in both denominators, a phenomenon that will continue to be observed for rebit-related results.

In Tables 1, 2, and 3, we present our conclusions, based on such high-precision calculations, as to the rational values () assumed by these induced-measure separability probabilities. Let us note that with the sole exception of , the rational values assumed by the (standard) two-qubit () induced states have both smaller denominators and numerators than the other two cases tabulated, indicative, presumably, in some manner, of their greater “naturalness.”

3. Three Companion Separability Probability Formulas

Further extending the entries of the two-qubit table (Table 2) but not explicitly showing the results here, to , application of the Mathematica command FindSequenceFunction to the sequence of length eighteen obtained plus subsequent simplification procedures yielded the governing ruleHere is the separability probability of the (15-dimensional) standard, complex two-qubit systems endowed with the induced measure . This formula, thus, successfully reproduces the entries of Table 2, as well as the subsequent ones () we have approximated to high precision, making use of the 11,401 moments in the Provost Legendre polynomial-based algorithm. For , formula (3) does, in fact, yield the apparent Hilbert-Schmidt separability probability of [7–9] (Table 2).

Similarly, employing a somewhat longer sequence , we obtained the quaternionic () counterpartyielding the (Hilbert-Schmidt) value of . Furthermore, for the rebit () scenario, making analogous use of the sequence , we foundyielding for , the result .

In Figure 1, we show a joint plot of these three separability probability formulas, with the rebit one () dominating the qubit one (), which in turn dominates the quaterbit () curve. In the limit , the three curves/probabilities all approach 1 (cf. [2]). We have found [15, Section III] through analytic means that, for each of and , as , the ratio of the logarithm of the st separability probability to the logarithm of the th separability probability is . Presumably, the pattern continues for larger , but the required computations have, so far, proved too challenging.

Figure 1 

Two-rebit two-qubit two-quaterbit separability probabilities, given by (5), (3), and (4), respectively, as functions of .

It is interesting to observe, additionally, that, for (i.e., ), a value not apparently susceptible to use of the principal -hypergeometric determinantal moment formula and the density approximation (inverse) procedure of Provost [17], the three basic formulas yield the (now smaller than Hilbert-Schmidt) further simple rational values , and for the rebit, qubit, and quaterbit cases, respectively (cf. [2, page 130]). Furthermore, for , the rebit formula has a singularity, the qubit formula yields 0, and the quaterbit one gives .

We have been able to formally extend this series of three formulas to other values of as well including obtaining similarly structured (increasingly larger) formulas. A major challenge that we are continuing to address is to find a single master formula that encompasses these several results and can itself yield the formula for any specific half-integer or integer value of (Appendix A).

4. Division of Separability Probabilities Based on Determinantal Inequalities

We have also begun to investigate related aspects of the geometry of random-induced generalized two-qubit states, making use of a second hypergeometric-based determinantal moment formula [18, Section II]:The range of the determinant difference variable is , and we shall approximate the contributions over to the total separability probabilities given in Tables 1, 2, and 3.

In [18], employing the first 9,451 of these moments (having set to zero) in the density approximation procedure of Provost [17], we obtained highly convincing numerical evidence that the basic set of three Hilbert-Schmidt separability probabilities () was evenly (symmetrically) split (i.e., ) between the two scenarios and . Now, with the use of 14,051 such determinantal moments, with , , we obtained an approximant equal to eight decimal places to for the case . Employing the total separability probability result of in Table 2, we find a complementary (larger) approximant of ; so the symmetry present in the Hilbert-Schmidt case (e.g., ) is lost for .

Similarly, for the , counterpart, we obtain an approximant equal, to almost twelve decimal places, to , when , and thus, for the complementary (larger) approximant.

To complete the basic triad, that is, and (for which convergence is typically weakest), for , we have an approximant equal, to more than six decimal places, to and a complementary (larger, again) approximant of . Note, again, the occurrence of powers of 2 in the case.

For , , it is interesting to note that the approximation of the probability is to ten decimal places. This is the same rational value we found above for the total separability probability. It, thus, appears that we can conclude that the complementary probability (i.e., for ) is now smaller, in fact, zero, in contrast to the case. The complementary probability also appears to be zero for the two companion cases of , , and .

In Figure 2, we show, based on numerical results using 9,201 moments, the proportion of the three basic total random induced separability probabilities (Tables 1, 2, and 3), as a function of , accounted for by the region . We have been investigating the possibility of obtaining explicit formulas, as we have been able to do above ((3), (4), and (5)) for the total separability probabilities (i.e., independently of whether or ), for these sets of complementary probabilities. To even hope to achieve such a goal, it seems necessary to fill in considerably more rows of Table 4 than we have so far been able to do (cf. [15]).

Figure 2 

Proportion of the total random-induced separability probabilities, based on 9,201 moments, accounted for by the region . The two-rebit () curve is dominant, the two-qubit () is intermediate, and the two-quaterbit () curve is subordinate.

5. Alternative Density Approximation Procedure

In pursuit of such a goal, we have developed an alternative (Appendix B) to the Legendre polynomial-based density approximation procedure of Provost [17], of which we have made abundant use above and in our earlier work [7, 8, 18]. Though well-conditioned, it perhaps is relatively slow to converge for our purposes, since it is taken as the baseline (starting) distribution, the uniform one, which is far from the sharply-peaked ones, with vanishing endpoints, we have encountered in our separability probability investigations. The approach outlined in Appendix B uses base functions of the form where is a small positive integer. Provost does present a number of codes other than the Legendre polynomial one including one based on Jacobi polynomials [17, pages 15, 24]. It employs an adaptive strategy of matching the first and second given moments to those of a beta distribution; but we have found this algorithm to be highly ill-conditioned in our specific applications.

6. Conclusions

We have reported above some considerable successes in our effort to extend to random induced measures [5, 6], earlier separability probability work [7, 8] based on the Hilbert-Schmidt measure (the particular symmetric case of the random induced measures), and the inequality . Further efforts using the more restrictive inequality utilized in Section 4 have been given in a subsequent paper [15]. These equivalences between certain hypergeometric-based formulas and difference equations have been noted.

Let us importantly note that in the recent study [19] the (random induced measure) separability probability problems posed above, have, in fact, been exhaustively formally solved for the “toy” seven-dimensional -states model [20] of density matrices. Here, contrastingly, we have concentrated on the more general cases of density matrices with none of the off-diagonal entries a priori nullified (as they are in the -states model). Although we have developed certain formulas here, for which the evidentiary support is quite considerable, we still lack formal proofs in this higher-dimensional venue.

We continue to investigate these problems in search of a still more definitive (“master formula”) resolution of them (Appendix A). As a possible tool in such research, we have developed (Appendix B), an alternative density approximation procedure to that of Provost [17], on which we have strongly relied to this point in obtaining exact separability probability results.

Appendices

A. Master Formula Investigation

This appendix is based on the random induced measure separability probability formulas we have obtained for .

The purpose is to find with respect to the normalized measure with parameter . The values correspond to the real, complex, quaternion cases, respectively. The obtained formulas have the formDefine

The first observation is that when is integer or half-integer is a rational function of , that is, a ratio of polynomials.

The second observation is that when is an integer thenwhere is a polynomial of degree with leading coefficient , and can be factored as , where is irreducible in general; furthermore The sequence of values for is

These conjectures imply that the degree of is

The coefficient of in (note that this is monic, coefficient of is ) is given byequivalentlyTo determine the second coefficient of , note that the second coefficient of is , so the second coefficient of is subtracted from . This coefficient is The second coefficient of is ; the sequence of values for is

Denote the coefficient of in by , and then from the calculated values () we find for that

The third observation is that when is a half-integer thenwhere is a polynomial of degree with leading coefficient .

B. A Modification of the Provost-Legendre Method Using Gegenbauer Polynomials

We consider the problem of approximating a density function with given moments using Jacobi polynomials for some choice of parameters. The technique uses a construction of Provost [17, Section 4] which is adapted for a specific aspect of the unknown probability density, namely, .

Suppose the density is supported on the interval with given (i.e., computable) moments , and is a family of orthogonal polynomials with weight function also on ; the structure constants areThe aim is to (implicitly) determine the expansionand to apply it to the approximation of (where )By orthogonality, for To evaluate the left hand side compute the coefficients in the expansionswhen are shifted Jacobi polynomials (this requires extra computation since the shortest expansions are in powers of or ); then

The main problem is to approximate for some given : soComputeand then and now we observe that the sum over is the coefficient of inTruncate the infinite series to obtain an approximation.

Jacobi Polynomials. We start with background information about general parameters and then specialize to equal parameters. The family is orthogonal for ; considerEquation (B.11) is from [21, ]. To shift to the interval set and the key quantities are found by and .