A confluence of numerical and theoretical results leads us to conjecture that the Hilbert–Schmidt separability probabilities of the 15- and 9-dimensional convex sets of complex and real two-qubit states (representable by 4 × 4 density matrices ρ) are and , respectively. Central to our reasoning are the modifications of two ansätze, recently advanced by Slater (2007 Phys. Rev. A 75 032326), involving incomplete beta functions B ν ( a , b ), where . We, now, set the separability function proportional to . Then, in the complex case—conforming to a pattern we find, manifesting the Dyson indices (β = 1, 2, 4) of random matrix theory—we take proportional to . We also investigate the real and complex qubit– qutrit cases. Now, there are two variables, , but they appear to remarkably coalesce into the product , so that the real and complex separability functions are again univariate in nature.
A confluence of numerical and theoretical results leads us to conjecture that the Hilbert–Schmidt separability probabilities of the 15- and 9-dimensional convex sets of complex and real two-qubit states (representable by 4 × 4 density matrices ρ) are and , respectively. Central to our reasoning are the modifications of two ansätze, recently advanced by Slater (2007 Phys. Rev. A 75 032326), involving incomplete beta functions B ν ( a , b ), where . We, now, set the separability function proportional to . Then, in the complex case—conforming to a pattern we find, manifesting the Dyson indices (β = 1, 2, 4) of random matrix theory—we take proportional to . We also investigate the real and complex qubit– qutrit cases. Now, there are two variables, , but they appear to remarkably coalesce into the product , so that the real and complex separability functions are again univariate in nature.
A confluence of numerical and theoretical results leads us to conjecture that the Hilbert-Schmidt separability probabilities of the 15- and 9-dimensional convex sets of complex and real two-qubit states (representable by 4 x 4 density matrices rho) are 8/33 and 8/17, respectively. Central to our reasoning are the modifications of two ansatze, recently advanced (quant-ph/0609006), involving incomplete beta functions B_{nu}(a,b), where nu= (rho_{11} rho_{44})/(rho_{22} rho_{33}). We, now, set the separability function S_{real}(nu) propto B_{nu}(nu,1/2},2) =(2/3) (3-nu) sqrt{nu}. Then, in the complex case -- conforming to a pattern we find, manifesting the Dyson indices (1, 2, 4) of random matrix theory-- we take S_{complex}(nu) propto S_{real}^{2} (nu). We also investigate the real and complex qubit-qutrit cases. Now, there are two Bloore ratio variables, nu_{1}= (rho_{11} rho_{55})(rho_{22} rho_{44}), nu_{2}= (rho_{22} rho_{66})(rho_{33} rho_{55}), but they appear to remarkably coalesce into the product, eta = nu_1 nu_2 = rho_{11} \rho_{66}}{\rho_{33} \rho_{44}}, so that the real and complex separability functions are again univariate in nature.
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.
Due to recent important work of \ifmmode \dot{Z}\else \.{Z}\fi{}yczkowski and Sommers [J. Phys. A 36, 10115 (2003); 36, 10083 (2003)], exact formulas are available (in terms of both the Hilbert-Schmidt and the Bures metrics) for the $({n}^{2}\ensuremath{-}1)$-dimensional and $[n(n\ensuremath{-}1)∕2\ensuremath{-}1]$-dimensional volumes of the complex and real $n\ifmmode\times\else\texttimes\fi{}n$ density matrices. However, no comparable formulas are available for the volumes (and, hence, probabilities) of various separable subsets of them. We seek to clarify this situation for the Hilbert-Schmidt metric for the simplest possible case of n=4, that is, two-qubit systems. Making use of the density matrix $(\ensuremath{\rho})$ parametrization of Bloore [J. Phys. A 9, 2059 (1976)], we are able to reduce each of the real and complex volume problems to the calculation of a one-dimensional integral, the single relevant variable being a certain ratio of diagonal entries, $\ensuremath{\nu}={\ensuremath{\rho}}_{11}{\ensuremath{\rho}}_{44}∕{\ensuremath{\rho}}_{22}{\ensuremath{\rho}}_{33}$. The associated integrand in each case is the product of a known Jacobian (highly oscillatory near $\ensuremath{\nu}=1$) and a certain unknown univariate function, which our extensive numerical (quasi Monte Carlo) computations indicate is very closely proportional to an (incomplete) Beta function ${B}_{\ensuremath{\nu}}(a,b)$, with a=1∕2, b=3 in the real case, and a=26∕5,b=3∕2 in the complex case. Assuming the full applicability of these specific incomplete Beta functions, we undertake separable-volume calculations.
We conduct a study based on the Bures (minimal monotone) metric, analogous to that recently reported for the Hilbert-Schmidt (flat or Euclidean) metric (arXiv:0704.3723v2). Among the interesting results obtained there had been proportionalities--in exact correspondence to the Dyson indices β= 1, 2, 4 of random matrix theory--between the fourth, second and first powers of the separability functions S_{type}(μ) for real, complex and quaternionic qubit-qubit scenarios, Here μ=\sqrt{\frac{ρ_{11} ρ_{44}}{ρ_{22} ρ_{33}}}, with ρbeing a 4 x 4 density matrix. Separability functions have proved useful--in the framework of the Bloore (correlation coefficient/off-diagonal scaling) parameterization of density matrices--for the calculation of separability probabilities. We find--for certain, basic simple scenarios (in which the diagonal entries of ρare unrestricted, and one or two off-diagonal [real, complex or quaternionic] pairs of entries are nonzero) --that these proportionalities no longer strictly hold in the Bures case, but do come remarkably close to holding.
We list in increasing order -- 1/3, 3/8, 2/5, 135 pi/1024, 16/(3 pi^2), 3 pi/16, 5/8, 105 pi/512, 2 - 435 pi/1024, 11/16, 1 -- a number of exact two-qubit Hilbert-Schmidt (HS) separability probabilities, we are able to compute. Each probability corresponds to a specific scenario -- a class of 4 x 4 density matrices (rho) with an m-subset (m<6) of its six off-diagonal pairs of symmetrically located entries set to zero. Initially, we consider only scenarios in which the (6-m) non-nullified pairs are real, but then permit them to be complex (as well as quaternionic) in nature also. The general analytical strategy we implement is based on the Bloore density matrix parameterization (J. Phys. A, 9, 2059 [1976]), allowing us to conveniently reduce the dimensionalities of required integrations. For each scenario, we identify a certain univariate ``separability function'' S_{scenario}(nu), where nu = rho_{11} rho_{44}/(rho_{22} rho_{33}). The integral over nu in [0,infty] of the product of this function with a scenario-specific (marginal) jacobian function J_{scenario}(nu) yields the HS {separable} volume (V^{HS}_{sep}). The ratio of V^{HS}_{sep} to the HS total (entangled and non-entangled) volume, gives us the HS scenario-specific separability probability. Among the possible forms that we have so far determined for S_{scenario}(nu) are piecewise combinations of c, c sqrt{nu} and c nu for nu in [0,1] and (in a dual manner) c, c/sqrt{nu} and c/nu for nu in [1,infty]. We also obtain bivariate separability functions S_{scenario)^{6 x 6}(nu_{1},nu_{2}} in the qubit-qutrit case, involving 6 x 6 density matrices, with ratio variables, nu_{1}= rho_{11} rho_{55}/(rho_{22} rho_{44}), and nu_{2}= rho_{22} rho_{66}/(rho_{33} rho_{55}). Additionally, we investigate parallel two-qutrit and three-qubit problems.
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