Compelling evidence-though yet no formal proof--has been adduced that the probability that a generic two-qubit state ($ρ$) is separable is $\frac{8}{33}$ (arXiv:1301.6617, arXiv:1109.2560, arXiv:0704.3723). Proceeding in related analytical frameworks, using a further determinantal moment formula of C. Dunkl (Appendix), we reach the conclusion that one-half of this probability arises when the determinantal inequality $|ρ^{PT}|>|ρ|$, where $PT$ denotes the partial transpose, is satisfied, and, the other half, when $|ρ|>|ρ^{PT}|$. These probabilities are taken with respect to the flat, Hilbert-Schmidt measure on the fifteen-dimensional convex set of $4 \times 4$ density matrices. We find fully parallel bisection/equipartition results for the previously adduced, as well, two-"re[al]bit" and two-"quater[nionic]bit"separability probabilities of $\frac{29}{64}$ and $\frac{26}{323}$, respectively. The computational results reported lend strong support to those obtained earlier--including the "concise formula" that yields them--most conspicuously amongst those findings being the $\frac{29}{64}, \frac{8}{33}$ and $\frac{26}{323}$ probabilities noted.
Compelling evidence-though yet no formal proof--has been adduced that the probability that a generic two-qubit state (ρ) is separable is 338 (arXiv:1301.6617, arXiv:1109.2560, arXiv:0704.3723). Proceeding in related analytical frameworks, using a further determinantal moment formula of C. Dunkl (Appendix), we reach the conclusion that one-half of this probability arises when the determinantal inequality ∣ρPT∣>∣ρ∣, where PT denotes the partial transpose, is satisfied, and, the other half, when ∣ρ∣>∣ρPT∣. These probabilities are taken with respect to the flat, Hilbert-Schmidt measure on the fifteen-dimensional convex set of 4×4 density matrices. We find fully parallel bisection/equipartition results for the previously adduced, as well, two-"re[al]bit" and two-"quater[nionic]bit"separability probabilities of 6429 and 32326, respectively. The computational results reported lend strong support to those obtained earlier--including the "concise formula" that yields them--most conspicuously amongst those findings being the 6429,338 and 32326 probabilities noted.
We report a concise answer--in the case of 2 x 2 systems--to the fundamental quantum-information-theoretic question as to "the volume of separable states" posed by Zyczkowski, Horodecki, Sanpera and Lewenstein (Phys. Rev. A, 58, 883 [1998]). We proceed by applying the Mathematica command FindSequenceFunction to a series of conjectured Hilbert-Schmidt generic 2 x 2 (rational-valued) separability probabilities p(a), a = 1, 2,...,32, with a = 1 indexing standard two-qubit systems, and a = 2, two-quater(nionic)bit systems. These 32 inputted values of p(a)--as well as 32 companion non-inputted values for the half-integers, a = 1/2 (two-re[al]bit) systems), 3/2,..., 63/2, are advanced on the basis of high-precision probability-distribution-reconstruction computations, employing 7,501 determinantal moments of partially transposed 4 x 4 density matrices. The function P(a) given by application of the command fully reproduces both of these 32-length sequences, and an equivalent outcome is obtained if the half-integral series is the one inputted. The lengthy expression (containing six hypergeometric functions) obtained for P(a) is, then, impressively condensed (by Qing-Hu Hou and colleagues), using Zeilberger's algorithm. For generic (9-dimensional) two-rebit systems, P(1/2) = 29/64, (15-dimensional) two-qubit systems, P(1) = 8/33, (27-dimensional) two-quaterbit systems, P(2) = 26/323, while for generic classical (3-dimensional) systems, P(0)=1.
Milz and Strunz recently reported substantial evidence to further support the previously conjectured separability probability of 338 for two-qubit systems (ρ) endowed with Hilbert-Schmidt measure. Additionally, they found that along the radius (r) of the Bloch ball representing either of the two single-qubit subsystems, this value appeared constant (but jumping to unity at the locus of the pure states, r=1). Further, they also observed (personal communication) such separability probability r-invariance, when using, more broadly, random induced measure (K=3,4,5,…), with K=4 corresponding to the (symmetric) Hilbert-Schmidt case. Among the findings here is that this invariance is maintained even after splitting the separability probabilities into those parts arising from the determinantal inequality ∣ρPT∣>∣ρ∣ and those from ∣ρ∣>∣ρPT∣>0, where the partial transpose is indicated. The nine-dimensional set of generic two-re[al]bit states endowed with Hilbert-Schmidt measure is also examined, with similar r-invariance conclusions. Contrastingly, two-qubit separability probabilities based on the Bures (minimal monotone) measure {\it diminish} with r. Moreover, we study the forms that the separability probabilities take as joint (bivariate) functions of the radii (rA,rB) of the Bloch balls of {\it both} single-qubit subsystems. Here, a form of Bloch radii {\it repulsion} for separable two-qubit systems emerges in {\it all} our several analyses. Separability probabilities tend to be smaller when the lengths of the two radii are closer. In Appendix A, we report certain companion analytic results for the much-investigated, more amenable (7-dimensional) X-states model.
Two-qubit X-matrices have been the subject of considerable recent attention, as they lend themselves more readily to analytical investigations than two-qubit density matrices of arbitrary nature. Here, we maximally exploit this relative ease of analysis to formally derive an exhaustive collection of results pertaining to the separability probabilities of generalized two-qubit X-matrices endowed with Hilbert–Schmidt and, more broadly, induced measures. Further, the analytical results obtained exhibit interesting parallels to corresponding earlier (but, contrastingly, not yet fully rigorous) results for general two-qubit states — deduced on the basis of determinantal moment formulas. Geometric interpretations can be given to arbitrary positive values of the random-matrix Dyson-index-like parameter [Formula: see text] employed.
To begin, we find certain formulas Q(k,α)=G1k(α)G2k(α) , for k=-1,0,1,…,9 . These yield that part of the total separability probability, P(k,α) , for generalized (real, complex, quaternionic , etc.) two-qubit states endowed with random induced measure, for which the determinantal inequality ρPT>ρ holds. Here ρ denotes a 4×4 density matrix, obtained by tracing over the pure states in 4×(4+k) -dimensions, and ρPT denotes its partial transpose. Further, α is a Dyson-index-like parameter with α=1 for the standard (15-dimensional) convex set of (complex) two-qubit states. For k=0 , we obtain the previously reported Hilbert-Schmidt formulas, with Q(0,1/2)=29/128 (the real case), Q(0,1)=4/33 (the standard complex case), and Q(0,2)=13/323 (the quaternionic case), the three simply equalling P(0,α)/2 . The factors G2k(α) are sums of polynomial-weighted generalized hypergeometric functions pFp-1 , p≥7 , all with argument z=27/64=(3/4)3 . We find number-theoretic-based formulas for the upper ( uik ) and lower ( bik ) parameter sets of these functions and, then, equivalently express G2k(α) in terms of first-order difference equations. Applications of Zeilberger’s algorithm yield “concise” forms of Q(-1,α) , Q(1,α) , and Q(3,α) , parallel to the one obtained previously (Slater 2013) for P(0,α)=2Q(0,α) . For nonnegative half-integer and integer values of α , Q(k,α
We extend the findings and analyses of our two recent studies (Phys. Rev. A, 75, 032326 [2007] and arXiv:0704.3723) by, first, obtaining numerical estimates of the separability function based on the (Euclidean, flat) Hilbert-Schmidt (HS) metric for the 27-dimensional convex set of quaternionic two-qubit systems. The estimated function appears to be strongly consistent with our previously-formulated Dyson-index (beta = 1, 2, 4) ansatz, dictating that the quaternionic (beta=4) separability function should be exactly proportional to the square of the separability function for the 15-dimensional convex set of two-qubit complex (beta=2) systems, as well as the fourth power of the separability function for the 9-dimensional convex set of two-qubit real (beta=1) systems. In particular, we conclude that S_{quat}(mu) =(6/71)^2 ((3-mu^2) mu)^4 =(S_{complex}(mu))^2. Here, mu =(rho_{11} rho_{44}/(rho_{22} rho_{33})^(1/2), where rho is a 4 x 4 two-qubit density matrix. We can, thus, supplement (and fortify) our previous assertion that the HS separability probability of the two-qubit complex states is 8/33= 0.242424, by claiming that its quaternionic counterpart is 72442944/936239725 = 0.0773765. We also comment on and analyze the odd beta=1 and 3 cases.
Previously, a formula, incorporating a 5F4 hypergeometric function, for the Hilbert-Schmidt-averaged determinantal moments ρPTnρk/ρk of 4×4 density-matrices ( ρ ) and their partial transposes ( |ρPT| ), was applied with k=0 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 ( k=1,2,… ) are found—viaa high-precision density approximation procedure—to assume interesting, relatively simple rational values in the two-re[al]bit ( α=1/2 ), (standard) two-qubit ( α=1 ), and two-quater[nionic]bit ( α=2 ) cases. We deduce rather simple companion (rebit, qubit, quaterbit, …) formulas that successfully reproduce the rational values assumed forgeneral k . These formulas are observed to share certain features, possibly allowing them to be incorporated into a single master formula.