A priori probability that two qubits are unentangled

In a previous study (quant-ph/9911058), several remarkably simple exact results were found, in certain specialized m-dimensional scenarios (m<5), for the a priori probability that a pair of qubits is unentangled/separable. The measure used was the volume element of the Bures metric (identically one-fourth the statistical distinguishability [SD] metric). Here, making use of a newly-developed (Euler angle) parameterization of the 4 x 4 density matrices (math-ph/0202002), we extend the analysis to the complete 15-dimensional convex set (C) of arbitrarily paired qubits -- the total SD volume of which is known to be \pi^8 / 1680 = \pi^8 / (2^4 3 5 7) = 5.64794. Using advanced quasi-Monte Carlo procedures (scrambled Halton sequences) for numerical integration in this high-dimensional space, we approximately (5.64851) reproduce that value, while obtaining an estimate of .416302 for the SD volume of separable states. We conjecture that this is but an approximation to \pi^6 /2310 = \pi^6 / (2 3 5 7 11) = .416186. The ratio of the two volumes, 8 / (11 \pi^2) = .0736881, would then constitute the exact Bures/SD probability of separability. The SD area of the 14-dimensional boundary of C is 142 \pi^7 / 12285 = 142 \pi^7 /(3^3 5 7 13) = 34.911, while we obtain a numerical estimate of 1.75414 for the SD area of the boundary of separable states.Comment: seven pages, LaTeX, minor revisions, to appear in Quantum Information Processin

Hypergeometric/Difference-Equation-Based Separability Probability Formulas and Their Asymptotics for Generalized Two-Qubit States Endowed with Random Induced Measure

We find equivalent hypergeometric- and difference-equation-based formulas, $Q(k,\alpha)= G_1^k(\alpha) G_2^k(\alpha)$, for $k = -1, 0, 1,\ldots,9$, 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 $|\rho^{PT}| >|\rho|$ holds. Here $\rho$ denotes a $4 \times 4$ density matrix and $\rho^{PT}$, its partial transpose, while $\alpha$ is a Dyson-index-like parameter with $\alpha = 1$ for the standard (15-dimensional) convex set of two-qubit states. The dimension of the space in which these density matrices is embedded is $4 \times (4 +k)$. For the symmetric case of $k=0$, we obtain the previously reported Hilbert-Schmidt formulas, with (the two-re[al]bit case) $Q(0,\frac{1}{2}) = \frac{29}{128}$, (the standard two-qubit case) $Q(0,1)=\frac{4}{33}$, and (the two-quater[nionic]bit case) $Q(0,2)= \frac{13}{323}$. The factors $G_2^k(\alpha)$ can be written as the sum of weighted hypergeometric functions $_{p}F_{p-1}$, $p \geq 7$, all with argument $\frac{27}{64} =(\frac{3}{4})^3$. We find formulas for the upper and lower parameter sets of these functions and, then, equivalently express $G_2^k(\alpha)$ in terms of first-order difference equations. The factors $G_1^k(\alpha)$ are equal to $(\frac{27}{64})^{\alpha-1}$ times ratios of products of six Pochhammer symbols involving the indicated parameters. Some remarkable $\alpha-$ and $k$-specific invariant asymptotic properties (again, involving $\frac{27}{64}$ and related quantities) of separability probability formulas emerge.Comment: 20 pages, 11 figure

Analytic Fits to Separable Volumes and Probabilities for Qubit-Qubit and Qubit-Qutrit Systems

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 \rho (math-ph/0202002) -- would be the products of 12-dimensional Haar measure \mu_{Haar} (common to each metric) and 3-dimensional measures \mu_{metric} (specific to each metric) over the 3d-simplex formed by the four eigenvalues of \rho. We attempt here to estimate/determine the 3-dimensional integrands (the products of the various [known] \mu_{metric}'s and an unknown symmetric weighting function W) remaining after the (putative) 12-fold integration of \mu_{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 \rho, we make extensive use of the Bloore parameterization (J. Phys. A 9 [1976], 2059) in a companion set of two-qubit separability analyses.Comment: 16 pages, modest revisions, retitled, some new references incorporated, including to quant-ph/0609006, "Two-Qubit Separability Probabilities and Beta Functions

Monotonicity Properties of Certain Measures over the Two-Level Quantum Systems

We demonstrate --- using the case of the two-dimensional quantum systems --- that the "natural measure on the space of density matrices describing N-dimensional quantum systems" proposed by Zyczkowski et al (quant-ph/9804024) does not belong to the class of normalized volume elements of monotone metrics on the quantum systems. Such metrics possess the statistically important property of being decreasing under stochastic mappings (coarse-grainings). We do note that the proposed natural measure (and certain evident variations upon it) exhibit quite specific monotonicity properties, but not of the form required for membership in the class.Comment: 4 pages, LaTeX. minor change

Bayesian Thermostatistical Analyses of Two-Level Complex and Quaternionic Systems

The three and five-dimensional convex sets of two-level complex and quaternionic quantum systems are studied in the Bayesian thermostatistical framework introduced by Lavenda. Associated with a given parameterization of each such set is a quantum Fisher (Helstrom) information matrix. The square root of its determinant (adopting an ansatz of Harold Jeffreys) provides a reparameterization-invariant prior measure over the set. Both such measures can be properly normalized and their univariate marginal probability distributions (which serve as structure functions) obtained. Gibbs (posterior) probability distributions can then be found, using Poisson's integral representation of the modified spherical Bessel functions. The square roots of the (classical) Fisher information of these Gibbs distributions yield (unnormalized) priors over the inverse temperature parameters.Comment: 6 pages, LaTeX, 6 postscript figures. This paper was submitted (in a non-LaTeX form) for publication in Dec., 1995, but not accepted. I have recast it into LaTeX for wider availability, primarily because it appears to have anticipated the recently-presented "quantum canonical ensemble" of Brody and Hughston (quant-ph/9709048

Quantum Statistical Thermodynamics of Two-Level Systems

We study four distinct families of Gibbs canonical distributions defined on the standard complex, quaternionic, real and classical (nonquantum) two-level systems. The structure function or density of states for any two-level system is a simple power (1, 3, 0 or -1) of the length of its polarization vector, while the magnitude of the energy of the system, in all four cases, is the negative of the logarithm of the determinant of the corresponding two-dimensional density matrix. Functional relationships (proportional to ratios of gamma functions) are found between the average polarizations with respect to the Gibbs distributions and the effective polarization temperature parameters. In the standard complex case, this yields an interesting alternative, meeting certain probabilistic requirements recently set forth by Lavenda, to the more conventional (hyperbolic tangent) Brillouin function of paramagnetism (which, Lavenda argues, fails to meet such specifications).Comment: 21 pages, LaTeX, 6 postscript figures. We show (in the concluding section) that the exponent for the power law behavior of the order parameter (twice the average polarization minus 1) for our (standard complex) alternative (proportional to a ratio of gamma functions) to the hyperbolic tangent Brillouin function of paramagnetism, equals one-half, being the same in both these case

Dyson-Index-Like Behavior of Bures Separability Functions

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 \beta = 1, 2, 4 of random matrix theory--between the fourth, second and first powers of the separability functions S_{type}(\mu) for real, complex and quaternionic qubit-qubit scenarios, Here \mu=\sqrt{\frac{\rho_{11} \rho_{44}}{\rho_{22} \rho_{33}}}, with \rho being 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 \rho are 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.Comment: 13 pages, 2 figure

Two-Qubit Rational-Valued Entanglement-Boundary Probability Densities and a Fisher Information Equality Conjecture

We consider a pair of one-parameter (alpha) families of generalized two-qubit determinantal Hilbert-Schmidt probability distributions, p_{alpha}(|rho^{PT}|) and q_{alpha}(|rho|), where rho is a 4 x 4 density matrix, rho^{PT}, its partial transpose, with |rho^{PT}| \in [-1/16,1/256] and |rho| \in [0, 1/256]. The Dyson-index-like (random matrix) parameter alpha is 1/2 for the 9-dimensional generic two-rebit systems, 1 for the 15-dimensional generic two-qubit systems,... Numerical (moment-based probability-distribution-reconstruction) analyses suggest the conjecture that the Fisher information--a measure over alpha--is identical for the two distinct families. Further, we study the values of p_{alpha}(0), the probability densities at the separability-entanglement boundary, with evidence strongly indicating that p_2(0) =7425/34 and p_3(0)= 7696/69. Despite extensive results of such a nature, we have not yet succeeded--in contrast with the corresponding rational-valued separability probabilities (arXiv.org:1301.6617)--in generating an underlying, explanatory ("concise") formula for p_{\alpha}(0), even though the corresponding denominators of both sets of rational-valued results are closely related, having almost identical (small) prime factors. The first derivatives p_{alpha}^{'}(0) are positive for alpha = 1/2 and 1, but negative for alpha > 1.Comment: 15 pages, 14 figures, 2 tables. Further analyses included. Substantial expansio

Self-duality, four-forms, and the eight-dimensional Yang-Mills/Dittmann-Bures field over the three-level quantum systems

Utilizing a number of results of Dittmann, we investigate the nature of the Yang-Mills field over the eight-dimensional convex set, endowed with the Bures metric, of three-level quantum systems. Parallelling the decomposition of eight-dimensional Euclidean fields by Corrigan, Devchand, Fairlie and Nuyts, as well as Figueoroa-O'Farrill and others, we investigate the properties of self-dual and anti-self-dual four-forms corresponding specifically to our Bures/non-Euclidean context. For any of a number of (nondegenerate) 3 x 3 density matrices, we are able to solve the eigenequation of the associated Hodge * operator with respect to the Bures metric. We obtain sets of (traceless) twenty-eight real eigenvalues, consisting of four singlets and three octets. The associated four-forms are found to exhibit quite simple behaviors, though we are not able to derive them in full generality.Comment: eleven pages, LaTeX, ten new figures, we substantially revise the paper, adding new analyses, while omitting much of the previous detail concerning the implementation of the (Euclidean) CDFN equations. This detail turned out to be largely irrelevant to the main (non-Euclidean) results of the pape

Two-Qubit Separabilities as Piecewise Continuous Functions of Maximal Concurrence. II--The Relevance of Dyson Indices

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.Comment: 5 pages, 3 figure