The Horodecki family employed the Jaynes maximum-entropy principle, fitting the mean (b(1)) of the Bell-CHSH observable (B). This model was extended by Rajagopal by incorporating the dispersion (sigma(2)(1)) of the observable, and by Canosa and Rossignoli, by generalizing the observable (B-alpha). We further extend the Horodecki one-parameter model in both these manners, obtaining a three-parameter (b(1), sigma(2)(1), alpha) two-qubit model, for which we find a highly interesting/intricate continuum (-infinity < a < infinity) of Hilbert-Schmidt (HS) separability probabilities-in which, the golden ratio is featured. Our model can be contrasted with the three-parameter (b(q), sigma(2)(q), q) one of Abe and Rajagopal, which employs a q(Tsallis)-parameter rather than a, and has simply q-invariant HS separability probabilities of 1/2 Our results emerge in a study initially focused on embedding certain information metrics over the two-level quantum systems into a q-framework. We find evidence, in this regard, that Srednicki's recently-stated biasedness criterion for noninformative priors yields rankings of priors fully consistent with an information-theoretic test of Clarke, previously applied to quantum systems by Slater.

Wu and Sprung (Phys. Rev. E, 48, 2595 (1993)) reproduced the first 500 nontrivial Riemann zeros, using a one-dimensional local potential model. They concluded - as did van Zyl and Hutchinson (Phys. Rev. E, 67, 066211 (2003)) - that the potential possesses a fractal structure of dimension d = 3/2. We model the nonsmooth fluctuating part of the potential by the alternating-sign sine series fractal of Berry and Lewis A(x, gamma). Setting d = 3/2, we estimate the frequency parameter (gamma), plus an overall scaling parameter (sigma) that we introduce. We search for that pair of parameters (gamma, sigma) that minimizes the least-squares fit S-n(gamma, sigma) of the lowest n eigenvalues - obtained by solving the one-dimensional stationary (nonfractal) Schrodinger equation with the trial potential (smooth plus nonsmooth parts) - to the lowest if Riemann zeros for n = 25. For the additional cases, we study, n = 50 and 75, we simply set sigma = 1. The fits obtained are compared to those found by using just the smooth part of the Wu-Sprung potential without any fractal supplementation. Some limited improvement - 5.7261 versus 6.39207 (n = 25), 11.2672 versus 11.7002 (n = 50), and 16.3119 versus 16.6809 (n = 75) - is found in our (nonoptimized, computationally bound) search procedures. The improvements are relatively strong in the vicinities of gamma = 3 and (its square) 9. Further, we extend the Wu-Sprung semiclassical framework to include higher order corrections from the Riemann-von Mangoldt formula (beyond the leading, dominant term) into the smooth potential.

Paralleling our recent computationally intensive (quasi-Monte Carlo) work for the case N=4 (e-print quant-ph/0308037), we undertake the task for N=6 of computing to high numerical accuracy, the formulas of Sommers and Zyczkowski (e-print quant-ph/0304041) for the (N-2-1)-dimensional volume and (N-2-2)-dimensional hyperarea of the (separable and nonseparable) NxN density matrices, based on the Bures (minimal monotone) metric-and also their analogous formulas (e-print quant-ph/0302197) for the (nonmonotone) flat Hilbert-Schmidt metric. With the same seven 10(9) well-distributed ("low-discrepancy") sample points, we estimate the unknown volumes and hyperareas based on five additional (monotone) metrics of interest, including the Kubo-Mori and Wigner-Yanase. Further, we estimate all of these seven volume and seven hyperarea (unknown) quantities when restricted to the separable density matrices. The ratios of separable volumes (hyperareas) to separable plus nonseparable volumes (hyperareas) yield estimates of the separability probabilities of generically rank-6 (rank-5) density matrices. The (rank-6) separability probabilities obtained based on the 35-dimensional volumes appear to be-independently of the metric (each of the seven inducing Haar measure) employed-twice as large as those (rank-5 ones) based on the 34-dimensional hyperareas. (An additional estimate-33.9982-of the ratio of the rank-6 Hilbert-Schmidt separability probability to the rank-4 one is quite clearly close to integral too.) The doubling relationship also appears to hold for the N=4 case for the Hilbert-Schmidt metric, but not the others. We fit simple exact formulas to our estimates of the Hilbert-Schmidt separable volumes and hyperareas in both the N=4 and N=6 cases.

The two principal/immediate influences-which we seek to interrelate here-upon the undertaking of this study are papers of Zyczkowski and Slomczynski [J. Phys. A 34, 6689 (2001)] and of Petz and Sudar [J. Math. Phys. 37, 2262 (1996)]. In the former work, a metric (the Monge one, specifically) over generalized Husimi distributions was employed to define a distance between two arbitrary density matrices. In the Petz-Sudar work (completing a program of Chentsov), the quantum analog of the (classically unique) Fisher information (monotone) metric of a probability simplex was extended to define an uncountable infinitude of Riemannian (also monotone) metrics on the set of positive definite density matrices. We pose here the questions of what is the specific/unique Fisher information metric for the (classically defined) Husimi distributions and how does it relate to the infinitude of (quantum) metrics over the density matrices of Petz and Sudar? We find a highly proximate (small relative entropy) relationship between the probability distribution (the quantum Jeffreys' prior) that yields quantum universal data compression, and that which (following Clarke and Barron) gives its classical counterpart. We also investigate the Fisher information metrics corresponding to the escort Husimi, positive-P and certain Gaussian probability distributions, as well as, in some sense, the discrete Wigner pseudoprobability. The comparative noninformativity of prior probability distributions-recently studied by Srednicki [Phys. Rev. A 71, 052107 (2005)]-formed by normalizing the volume elements of the various information metrics, is also discussed in our context. (c) 2006 American Institute of Physics.

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.) are 8/33 and 8/17, respectively. Central to our reasoning are the modifications of two ansatze, recently advanced by Slater (2007 Phys. Rev. A 75 032326), involving incomplete beta functions B.(a, b), where. v = rho(11)rho(44)/rho(22)rho(33). We, now, set the separability function S-real(v) proportional to Bv(v, 1/2, 2) = 2/3(3 -v)root v. Then, in the complex case - conforming to a pattern we find, manifesting the Dyson indices (beta = 1, 2, 4) of random matrix theory - we take S-complex(v) proportional to S-real(2)(v). We also investigate the real and complex qubit-qutrit cases. Now, there are two variables,v(1) =rho(11)rho(55)/rho(22)rho(44),v(2) =rho(22)rho(66)/rho(33)rho(55), but they appear to remarkably coalesce into the product. eta = v(1)v(2) = rho(11)rho(66)/rho(33)rho(44), so that the real and complex separability functions are again univariate in nature.

Extensive numerical integration results lead us to conjecture that the silver mean, that is, sigmaAg = root2 - 1 approximate to 0.414214 plays a fundamental role in certain geometries (those given by monotone metrics) imposable on the 15-dimensional convex set of two-qubit systems. For example, we hypothesize that the volume of separable two-qubit states, as measured in terms of (four times) the minimal monotone or Bures metric is sigma(Ag)/3, and 10sigma(Ag) in terms of (four times) the Kubo-Mori monotone metric. Also, we conjecture, in terms of (four times) the Bures metric, that part of the 14-dimensional boundary of separable states consisting generically of rank four 4 x 4 density matrices has volume ("hyperarea") 55sigma(Ag)/39, and that part composed of rank-three density matrices, 43sigma(Ag)/39, so the total boundary hyperarea would be 98sigma(Ag)/39. While the Bures probability of separability (approximate to0.07334) dominates that (approximate to0.050339) based on the Wigner-Yanase metric (and all other monotone metrics) for rank-four states, the Wigner-Yanase (approximate to0.18228) strongly dominates the Bures (approximate to0.03982) for the rank-three states. (C) 2004 Elsevier B.V. All rights reserved.