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

Abstract

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.