Quantum Large Deviations

TL;DR

This paper presents a new proof of the quantum analogue of Varadhan's Theorem using large deviation analysis and extends it to non-commuting variables in mean-field quantum spin systems, providing a variational formula.

Contribution

It introduces a novel proof technique for quantum large deviations and generalizes the variational formula to non-commuting quantum variables.

Findings

New proof of quantum Varadhan's Theorem using large deviation analysis

Extension of the theorem to non-commuting variables in quantum spin systems

Derivation of a variational formula for general mean-field quantum models

Abstract

We reconsider the quantum analogue of Varadhans Theorem proved by Petz, Raggio and Verbeure. They proved this theorem using standard techniques in quantum statistical mechanics of lattice systems to arrive at a variational formula over states on an operator algebra, which can subsequently be reduced to a variational formula in terms of a single real variable. In this paper a new proof is given using a quantum version of the large deviation analysis together with the Trotter product formula. The proof is subsequently extended to the general case of q non-commuting variables resulting in a variational formula for general mean-field quantum spin systems as first derived by Raggio and Werner.