Maximum Entropy Models for Quantum Systems

Andrzej Łuczak,Hanna Podsędkowska,Michał Seweryn

doi:10.3390/e19010001

Abstract

We show that for a finite von Neumann algebra, the states that maximise Segal’s entropy with a given energy level are Gibbs states. This is a counterpart of the classical result for the algebra of all bounded linear operators on a Hilbert space and von Neumann entropy.

Highlights

Let B(H) be the algebra of all bounded linear operators on a Hilbert space H, and let tr be the canonical trace on B(H)
The classical result says that the maximal value of the entropy for such states is attained for a so-called Gibbs state; βH
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

Summary

Introduction

Let B(H) be the algebra of all bounded linear operators on a Hilbert space H, and let tr be the canonical trace on B(H). For a state ρ on B(H) represented by a density matrix D, its von Neumann entropy is defined by. Let H be the Hamiltonian of a physical system whose (bounded) observables are represented by B(H). The expected value of the energy in the state ρ is given by ρ( H ) = tr DH. 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; βH that is, a state with the density matrix tre e βH for some β ∈ R. 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

Preliminaries and Notation

Maximum Segal’s Entropy States

Conclusions