Quantum Large Deviations
T. C. Dorlas
TL;DR
This work delivers a new proof of the quantum PRV large-deviation theorem (Petz–Petz–Raggio–Verbeure) for finite-dimensional, tensor-product algebras using a quantum large-deviation framework and the Trotter product formula. It identifies a cumulant generating function $C$ and its Legendre transform $I$ to obtain a sharp variational formula for the trace in the large-$n$ limit and extends the analysis from a single non-commuting pair to multi-variable settings, including two- and multi-variable generalizations and general mean-field quantum spin systems. The paper provides both upper and lower bounds that match, yields explicit rate functions, and demonstrates applicability to canonical models such as the mean-field transverse-field Ising and Heisenberg models, while supplying foundational lemmas (e.g., a non-commutative Hölder inequality) and a quantum stochastic-process construction. These results broaden the scope of quantum large deviations and furnish variational tools for analyzing mean-field quantum dynamics and thermodynamics. Furthermore, the appendix underpins the technical backbone with a rigorous non-commutative Hölder framework essential for the multi-variable bounds.
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.