Maximum Entropy Models for Quantum Systems

Let E, belonging to the spectrum of H, be a fixed energy level. We are interested in the states for which the expected value of the energy equals E; i.e., in the states ρ such that ρ(H) = E. The classical result says that the maximal value of the entropy for such states is attained for a so-called Gibbs state; that is, a state with the density matrix e βH tr eβH for some β ∈ R. In this note, we aim to show a similar result in the situation where B(H) is replaced by a finite von Neumann algebra, and the von Neumann entropy is replaced by Segal’s entropy.