Firstly, we reduce the long-standing problem of ascertaining the Hilbert-Schmidt probability that a generic pair of qubits is separable to that of determining the specific nature of a one-dimensional (separability) function of the radial coordinate (r) of the unit ball in 15-dimensional Euclidean space, and similarly for a generic pair of rebits, using the 9-dimensional unit ball. Separability probabilities, could, then, be directly obtained by integrating the products of these functions (which we numerically estimate and plot) with jacobian factors of r^m over r in [0,1], with m=17 for the two-rebit case, and m=29 in the two-qubit instance. Secondly, we repeat the analyses, but for the replacement of r as the free variable, by the azimuthal angle phi \in [0,2 pi]--with the associated jacobian factors now being, trivially, unity. So, the separability probabilities, then, become simply the areas under the curves generated. For our analyses, we employ an interesting Cholesky-decomposition parameterization of the 4 x 4 density matrices. Thirdly, in an exploratory investigation, we examine the Hilbert-Schmidt separability probability question using the three-dimensional visualization (cube/tetrahedron/octahedron) of two qubits associated with Avron, Bisker and Kenneth, and Leinaas, Myrheim and Ovrum, among others.
Firstly, we reduce the long-standing problem of ascertaining the Hilbert-Schmidt probability that a generic pair of qubits is separable to that of determining the specific nature of a one-dimensional (separability) function of the radial coordinate (r) of the unit ball in 15-dimensional Euclidean space, and similarly for a generic pair of rebits, using the 9-dimensional unit ball. Separability probabilities, could, then, be directly obtained by integrating the products of these functions (which we numerically estimate and plot) with jacobian factors of r^m over r in [0,1], with m=17 for the two-rebit case, and m=29 in the two-qubit instance. Secondly, we repeat the analyses, but for the replacement of r as the free variable, by the azimuthal angle phi \in [0,2 pi]--with the associated jacobian factors now being, trivially, unity. So, the separability probabilities, then, become simply the areas under the curves generated. For our analyses, we employ an interesting Cholesky-decomposition parameterization of the 4 x 4 density matrices. Thirdly, in an exploratory investigation, we examine the Hilbert-Schmidt separability probability question using the three-dimensional visualization (cube/tetrahedron/octahedron) of two qubits associated with Avron, Bisker and Kenneth, and Leinaas, Myrheim and Ovrum, among others.
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.
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 seek to derive the probability—expressed in terms of the Hilbert–Schmidt (Euclidean or flat) metric—that a generic (nine-dimensional) real two-qubit system is separable, by implementing the well-known Peres–Horodecki test on the partial transposes (PTs) of the associated 4 × 4 density matrices (ρ). But the full implementation of the test—requiring that the determinant of the PT be nonnegative for separability to hold—appears to be, at least presently, computationally intractable. So, we have previously implemented—using the auxiliary concept of a diagonal-entry-parameterized separability function (DESF)—the weaker implied test of nonnegativity of the six 2 × 2 principal minors of the PT. This yielded an exact upper bound on the separability probability of . Here, we piece together (reflection-symmetric) results obtained by requiring that each of the four 3 × 3 principal minors of the PT, in turn, be nonnegative, giving an improved/reduced upper bound of . Then, we conclude that a still further improved upper bound of can be found by similarly piecing together the (reflection-symmetric) results of enforcing the simultaneous nonnegativity of certain pairs of the four 3 × 3 principal minors. Numerical simulations—as opposed to exact symbolic calculations—indicate, on the other hand, that the true probability is certainly less than . Our analyses lead us to suggest a possible form for the true DESF, yielding a separability probability of , while the absolute separability probability of provides the best exact lower bound established so far. In deriving our improved upper bounds, we rely repeatedly upon the use of certain integrals over cubes that arise. Finally, we apply an independence assumption to a pair of DESFs that comes close to reproducing our numerical estimate of the true separability function.
We begin by advancing conjectures, closely fitting numerical estimates, for the Hilbert-Schmidt separability/PPT probabilities of the 35- and 63-dimensional convex sets of (2×3) qubit-qutrit and (2×4) qubit-qudit states, as well as certain related scenarios. We reason that the underlying volume formulas might closely parallel those now well-established--via a Lovas-Andai formula--in their lower (15)-dimensional two-qubit analogue. The qubit-qutrit conjecture we obtain is (103)3=23⋅5333=100027=0.027, with the counterpart two-qubit separability probablity being 338=3⋅1123≈0.242424. In the (2×4) qubit-qudit case, we advance 1237516=32⋅53⋅1142≈0.001292929 as the PPT-probability. The numerators of the probabilities for 2×m scenarios are, thus, possibly simply powers of m. We also search for the rebit-retrit counterpart to the established two-rebit result of 6429=2629≈0.453125. Additionally, we find that the Hilbert-Schmidt separability/PPT-probabilities for the two-rebit, rebit-retrit and two-retrit X-states all equal 3π216≈0.54038, as well as more generally, that the probabilities based on induced measures are equal across these three sets of X-states. Then, we extend the Lovas-Andai two-qubit framework from Hilbert-Schmidt measures to induced ones, raising the possibility of a still more general master formula.
Jakobczyk and Siennicki studied two-dimensional sections of a set of (generalized) Bloch vectors corresponding to n x n density matrices of two-qubit systems (that is, the case n = 4). They found essentially five different types of (nontrivial) separability regimes. We compute the Euclidean/Hilbert-Schmidt (HS) separability probabilities assigned to these regimes, and conduct parallel two-dimensional sectional analyses for the higher-level cases n=6,8,9 and 10. Making use of the newly-introduced capability for integration over implicitly defined regions of version 5.1 of Mathematica -- also fruitfully used in our n=4 three-parameter entropy-maximization-based study quant-ph/0507203 -- we obtain a wide-ranging variety of exact HS-probabilities. For n>6, the probabilities are those of having a partial positive transpose (PPT). For the n=6 case, we also obtain biseparability probabilities; in the n=8,9 instances, bi-PPT probabilities; and for n=8, tri-PPT probabilities. By far, the most frequently recorded probability for n>4 is π/4 = 0.785398. We also conduct a number of related analyses, pertaining to the (one-dimensional) boundaries (both exterior and interior) of the separability and PPT domains, and attempt (with limited success) some exact calculations pertaining to the 9-dimensional (real) and 15-dimensional (complex) convex sets of two-qubit density matrices -- for which HS-separability probabilities have been conjectured, but not verified.