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 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 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.
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.
We importantly amend a certain parenthetical remark made in Part I (arXiv:0806.3294), to the effect that although two-qubit diagonal-entry-parameterized separability functions had been shown (arXiv:0704.3723) to clearly conform to a pattern dictated by the "Dyson indices" (beta = 1 [real], 2 [complex], 4 [quaternionic]) of random matrix theory, this did not appear to be the case with regard to eigenvalue-parameterized separability functions (ESFs). But upon further examination of the extensive numerical analyses reported in Part I, we find quite convincing evidence that adherence to the Dyson-index pattern does also hold for ESFs, at least as regards the upper half-range 1/2 <= C <= 1 of the maximal concurrence over spectral orbits, C. To be specific, it strongly appears that in this upper half-range, the real two-qubit ESF is simply proportional to (2-2 C)^(3/2), and its complex counterpart--in conformity to the Dyson-index pattern--proportional to the square of the real ESF, that is, (2-2 C)^3. The previously documented piecewise continuous ("semilinear") behavior in the lower half-range still appears, however, to lack any particular Dyson-index-related interpretation.
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.
The probability that a generic real, complex or quaternionic two-qubit state is separable can be considered to be the sum of three contributions. One is from those states that are absolutely separable, that is those (which can not be entangled by unitary transformations) for which the maximal concurrence over spectral orbits ( C max ) is zero. The other two contributions are from the states for which , and for which . We have previously (arXiv:0805.0267) found exact formulas for the absolutely separable contributions in terms of the Hilbert–Schmidt metric over the quantum states, and here advance hypotheses as to the exact contributions for . A crucial element in understanding the two contributions for C max > 0 is the nature of the ratio ( R ) of the C max -parameterized separability function for the complex states to the square of the comparable function for the real states–both such functions having clearly displayed jump discontinuities at . For , the ratio R appears to be of the form 1 + kC max , except near , while for , there is strong numerical evidence that it equals 2 (thus, according to the Dyson-index pattern of random matrix theory). Related phenomena also occur for the minimally degenerate two-qubit states and the qubit–qutrit states. Our results have immediate application to the computation of separability probabilities in terms of other metrics, such as the Bures (minimal monotone) metric. The paper begins with continuous embeddings of the separability probability question in terms of four metrics of interest, using 'generalized Peres–Horodecki conditions'.
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.