The generic real (β = 1) and complex (β = 2) two-qubit states are 9-dimensional and 15-dimensional in nature, respectively. The total volumes of the spaces they occupy with respect to the Hilbert–Schmidt and Bures metrics are obtainable as special cases of formulae of Życzkowski and Sommers. We claim that if one could determine certain metric- independent three-dimensional ' eigenvalue -parameterized separability functions' (EPSFs), S (1,β) 4 (λ 1 ...λ 4 ), then these formulae could be readily modified so as to yield the Hilbert–Schmidt and Bures volumes occupied by only the separable two-qubit states (and hence associated separability probabilities ). Motivated by analogous earlier analyses of ' diagonal-entry -parameterized separability functions', we further explore the possibility that such three-dimensional EPSFs might, in turn, be expressible as univariate functions of some special relevant variable—which we hypothesize to be the maximal concurrence (0 ⩽ C ⩽ 1) over spectral orbits. Extensive numerical results that we obtain are rather closely supportive of this hypothesis. Both the real and complex estimated EPSFs exhibit clearly pronounced jumps of magnitude roughly 50% at , as well as a number of additional matching discontinuities.
The generic real (β = 1) and complex (β = 2) two-qubit states are 9-dimensional and 15-dimensional in nature, respectively. The total volumes of the spaces they occupy with respect to the Hilbert–Schmidt and Bures metrics are obtainable as special cases of formulae of Życzkowski and Sommers. We claim that if one could determine certain metric- independent three-dimensional ' eigenvalue -parameterized separability functions' (EPSFs), S (1,β) 4 (λ 1 ...λ 4 ), then these formulae could be readily modified so as to yield the Hilbert–Schmidt and Bures volumes occupied by only the separable two-qubit states (and hence associated separability probabilities ). Motivated by analogous earlier analyses of ' diagonal-entry -parameterized separability functions', we further explore the possibility that such three-dimensional EPSFs might, in turn, be expressible as univariate functions of some special relevant variable—which we hypothesize to be the maximal concurrence (0 ⩽ C ⩽ 1) over spectral orbits. Extensive numerical results that we obtain are rather closely supportive of this hypothesis. Both the real and complex estimated EPSFs exhibit clearly pronounced jumps of magnitude roughly 50% at , as well as a number of additional matching discontinuities.
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'.
We investigate the possibility of deriving analytical formulas for the 15-dimensional separable volumes, in terms of any of a number of metrics of interest (Hilbert-Schmidt [HS], Bures,...), of the two-qubit (four-level) systems. This would appear to require 15-fold symbolic integrations over a complicated convex body (defined by both separability and feasibility constraints). The associated 15-dimensional integrands -- in terms of the Tilma-Byrd-Sudarshan Euler-angle-based parameterization of the 4 x 4 density matrices ρ(math-ph/0202002) -- would be the products of 12-dimensional Haar measure μ_{Haar} (common to each metric) and 3-dimensional measures μ_{metric} (specific to each metric) over the 3d-simplex formed by the four eigenvalues of ρ. We attempt here to estimate/determine the 3-dimensional integrands (the products of the various [known] μ_{metric}'s and an unknown symmetric weighting function W) remaining after the (putative) 12-fold integration of μ_{Haar} over the twelve Euler angles. We do this by fitting W so that the conjectured HS separable volumes and hyperareas (quant-ph/0410238; cf. quant-ph/0609006) are reproduced. We further evaluate a number of possible choices of W by seeing how well they also yield the conjectured separable volumes for the Bures, Kubo-Mori, Wigner-Yanase and (arithmetic) average monotone metrics and the conjectured separable Bures hyperarea (quant-ph/0308037,Table VI). We, in fact, find two such exact (rather similar) choices that give these five conjectured (non-HS) values all within 5%. In addition to the above-mentioned Euler angle parameterization of ρ, we make extensive use of the Bloore parameterization (J. Phys. A 9 [1976], 2059) in a companion set of two-qubit separability analyses.
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.
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.