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
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 find equivalent hypergeometric- and difference-equation-based formulas, Q(k,α)=G1k(α)G2k(α), for k=−1,0,1,…,9, for that (rational-valued) portion of the total separability probability for generalized two-qubit states endowed with random induced measure, for which the determinantal inequality ∣ρPT∣>∣ρ∣ holds. Here ρ denotes a 4×4 density matrix and ρPT, its partial transpose, while α is a Dyson-index-like parameter with α=1 for the standard (15-dimensional) convex set of two-qubit states. The dimension of the space in which these density matrices is embedded is 4×(4+k). For the symmetric case of k=0, we obtain the previously reported Hilbert-Schmidt formulas, with (the two-re[al]bit case) Q(0,21)=12829, (the standard two-qubit case) Q(0,1)=334, and (the two-quater[nionic]bit case) Q(0,2)=32313. The factors G2k(α) can be written as the sum of weighted hypergeometric functions pFp−1, p≥7, all with argument 6427=(43)3. We find formulas for the upper and lower parameter sets of these functions and, then, equivalently express G2k(α) in terms of first-order difference equations. The factors G1k(α) are equal to (6427)α−1 times ratios of products of six Pochhammer symbols involving the indicated parameters. Some remarkable α− and k-specific invariant asymptotic properties (again, involving 6427 and related quantities) of separability probability formulas emerge.
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.
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.
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.
We report major advances in the research program initiated in 'Moment-based evidence for simple rational-valued Hilbert–Schmidt generic 2 × 2 separability probabilities' (Slater and Dunkl 2012 J. Phys. A: Math. Theor. 45 095305). A highly succinct separability probability function P (α) is put forth, yielding for generic (nine-dimensional) two-rebit systems, , (15-dimensional) two-qubit systems, and (27-dimensional) two-quater(nionic)bit systems, . This particular form of P (α) was obtained by Qing-Hu Hou by applying Zeilberger's algorithm ('creative telescoping') to a fully equivalent—but considerably more complicated—expression containing six 7 F 6 hypergeometric functions (all with argument ). That hypergeometric form itself had been obtained using systematic, high-accuracy probability-distribution-reconstruction computations. These employed 7501 determinantal moments of partially transposed 4 × 4 density matrices, parameterized by . From these computations, exact rational-valued separability probabilities were discernible. The (integral/half-integral) sequences of 32 rational values then served as input to the Mathematica FindSequenceFunction command, from which the initially obtained hypergeometric form of P (α) emerged.
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.