On local quantum Gibbs states

TL;DR

This paper investigates the problem of finding local quantum Gibbs states by minimizing quantum entropies under local constraints like density and energy, proving the existence and uniqueness of such states.

Contribution

It introduces a framework for local quantum entropy minimization with fixed local constraints and proves the existence and uniqueness of the minimizer.

Findings

Unique constrained minimizer of quantum entropy exists.

Method to handle lack of compactness in local energy constraint.

Auxiliary minimization problem aids in the proof.

Abstract

We address in this work the problem of minimizing quantum entropies under local constraints. We suppose macroscopic quantities such as the particle density, current, and kinetic energy are fixed at each point of $\Rm^d$, and look for a density operator over $L^2(\Rm^d)$ minimizing an entropy functional. Such minimizers are referred to as a local Gibbs states. This setting is in constrast with the classical problem of prescribing global constraints, where the total number of particles, total current, and total energy in the system are fixed. The question arises for instance in the derivation of fluid models from quantum dynamics. We prove, under fairly general conditions, that the entropy admits a unique constrained minimizer. Due to a lack of compactness, the main difficulty in the proof is to show that limits of minimizing sequences satisfy the local energy constraint. We tackle this issue by introducing a simpler auxiliary minimization problem and by using a monotonicity argument involving the entropy.