Volume Elements of Monotone Metrics on the n x n Density Matrices as Densities-of-States for Thermodynamic Purposes. II

Abstract

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.