Among the monotone metrics on the (n^{2} - 1)-dimensional convex set of n x n density matrices, as Petz and Sudar have recently elaborated, there are a minimal (Bures) and a maximal one. We examine the proposition that it is physically meaningful to treat the volume elements of these metrics as densities-of-states for thermodynamic purposes. In the n = 2 (spin-1/2) case, use of the maximal monotone metric, in fact, does lead to the adoption of the Langevin (and not the Brillouin) functions, thus, completely conforming with a recent probabilistic argument of Lavenda. Brody and Hughston also arrived at the Langevin function in an analysis based on the Fubini-Study metric. It is a matter of some interest, however, that in the first (subsequently modified) version of their paper, they had reported a different result, one fully consistent with the alternative use of the minimal monotone metric. In this part I of our investigation, we first study scenarios involving partially entangled spin-1/2 particles (n = 4, 6,...) and then a certain three-level extension of the two-level systems. In part II, we examine, in full generality, and with some limited analytical success, the cases n = 3 and 4.
Among the monotone metrics on the (n^{2} - 1)-dimensional convex set of n x n density matrices, as Petz and Sudar have recently elaborated, there are a minimal (Bures) and a maximal one. We examine the proposition that it is physically meaningful to treat the volume elements of these metrics as densities-of-states for thermodynamic purposes. In the n = 2 (spin-1/2) case, use of the maximal monotone metric, in fact, does lead to the adoption of the Langevin (and not the Brillouin) functions, thus, completely conforming with a recent probabilistic argument of Lavenda. Brody and Hughston also arrived at the Langevin function in an analysis based on the Fubini-Study metric. It is a matter of some interest, however, that in the first (subsequently modified) version of their paper, they had reported a different result, one fully consistent with the alternative use of the minimal monotone metric. In this part I of our investigation, we first study scenarios involving partially entangled spin-1/2 particles (n = 4, 6,...) and then a certain three-level extension of the two-level systems. In part II, we examine, in full generality, and with some limited analytical success, the cases n = 3 and 4.
Brody and Hughston and, independently, Slater have recently developed quantum-theoretic alternatives, based on certain Riemannian geometries, to the conventional (semiclassical/Jaynesian) density matrix approach to the canonical ensemble. In both of these studies, attention was focused on the example of a single spin-1/2 particle. In this communication, the particular line of argument previously employed by Slater (relying upon the minimal monotone or Bures metric, rather than the Fubini-Study metric utilized by Brody and Hughston) is extended to several forms of partial entanglement of two or more spin-1/2 particles. Parallel analyses are, then, conducted based on the maximal monotone metric, which for a single spin-1/2 particle yields the Langevin function. This function, to which Brody and Hughston were led in further work of theirs, accords, in contrast to the Brillouin function, with certain probabilistic requirements espoused by Lavenda.
We derive explicit expressions for the volume elements of both the minimal and maximal monotone metrics over the (n^{2} - 1)-dimensional convex set of n x n density matrices for the cases n = 3 and 4. We make further progress for the specific n = 3 maximal-monotone case, by taking the limit of a certain ratio of integration results, obtained using an orthogonal set of eight coordinates. By doing so, we find remarkably simple marginal probability distributions based on the corresponding volume element, which we then use for thermodynamic purposes. We, thus, find a spin-1 analogue of the Langevin function. In the fully general n = 4 situation, however, we are impeded in making similar progress by the inability to diagonalize a 3 x 3 Hermitian matrix and thereby obtain an orthogonal set of coordinates to use in the requisite integrations.
We study the interrelationships between the Fisher information metric recently introduced, on the basis of maximum entropy considerations, by Brody and Hughston (quant-ph/9906085) and the monotone metrics, as explicated by Petz and Sudar. This new metric turns out to be not strictly monotone in nature, and to yield (via its normalized volume element) a prior probability distribution over the Bloch ball of two-level quantum systems that is less noninformative than those obtained from any of the monotone metrics, even the minimal monotone (Bures) metric. We best approximate the additional information contained in the Brody-Hughston prior over that contained in the Bures prior by constructing a certain Bures posterior probability distribution. This is proportional to the product of the Bures prior and a likelihood function based on four pairs of spin measurements oriented along the diagonal axes of an inscribed cube.
We pursue a number of analytical directions, motivated to some extent initially by the possibility of developing a methodology for formally proving or disproving a certain conjecture of quantum-theoretical relevance (quant-ph/0308037). It asserts that the 15-dimensional volume occupied by the separable two-qubit density matrices is (\sqrt{2}-1)/3, as measured in terms of the statistical distinguishability metric (four times the Bures or minimal monotone metric). Somewhat disappointingly, however, the several various analyses that we report, though we hope of independent/autonomous interest, appear to provide small indication of how to definitively resolve the conjecture. Among our studies here are ones of: (1) the Bures volumes of the two-dimensional sections of Bloch vectors for a number of the Jakobczyk-Siennicki two-qubit scenarios; (2) the structure of certain convex polytopes of separable density matrices; and (3) the diagonalization of 15 x 15 Bures metric tensors.
Zyczkowski, Horodecki, Sanpera, and Lewenstein (ZHSL) recently proposed a ``natural measure'' on the N-dimensional quantum systems (quant-ph/9804024), but expressed surprise when it led them to conclude that for N = 2 x 2, disentangled (separable) systems are more probable (0.632) in nature than entangled ones. We contend, however, that ZHSL's (rejected) intuition has, in fact, a sound theoretical basis, and that the a priori probability of disentangled 2 x 2 systems should more properly be viewed as (considerably) less than 0.5. We arrive at this conclusion in two quite distinct ways, the first based on classical and the second, quantum considerations. Both approaches, however, replace (in whole or part) the ZHSL (product) measure by ones based on the volume elements of monotone metrics, which in the classical case amounts to adopting the Jeffreys' prior of Bayesian theory. Only the quantum-theoretic analysis (which yields the smallest probabilities of disentanglement) uses the minimum number of parameters possible, N^2 - 1, as opposed to N^2 + N - 1 (although this over-parameterization, as recently indicated by Byrd, should be avoidable). However, despite substantial computation, we are not able to obtain precise estimates of these probabilities, and the need for additional (possibly supercomputer) analyses is indicated (particularly so, for higher-dimensional quantum systems, such as the 2 x 3 we also study here).
We apply and extend recent results of Krattenthaler and Slater (quant-ph/9612043), who sought quantum analogs of seminal work on universal data compression of Clarke and Barron. KS obtained explicit formulas for the eigenvalues and eigenvectors of the 2^m x 2^m density matrices gotten by averaging the m-fold tensor products with themselves of the 2 x 2 density matrices. The weighting was done with respect to a one-parameter family of probability distributions, all the members of which are spherically-symmetric over the "Bloch sphere" of two-level quantum systems. This family includes the normalized volume element of the minimal monotone (Bures) metric. In this letter, we conduct parallel analyses (for m =2,3,4), based on a natural measure on the density matrices recently proposed by Zyczkowski, Horodecki, Sanpera and Lewenstein (quant-ph/9804024) and find interesting similarities and differences with the findings of KS. In addition, we are able to obtain exact analogous results, based on the measure of ZHSL, for the twofold tensor products of the 3 x 3 density matrices.