Quantum concentration inequalities

TL;DR

This paper develops quantum transportation cost inequalities using non-commutative geometric methods, applicable to Gibbs states, quantum Markov semigroups, and eigenvalue distributions, with implications for quantum thermodynamics.

Contribution

It introduces quantum extensions of classical transportation inequalities, establishing their validity for Gibbs states, quantum Markov processes, and eigenvalue distributions, with improved temperature ranges and constants.

Findings

High temperature Gibbs states satisfy a TCI with O(|V|) scaling.

TCI holds for fixed points of quantum Markov semigroups with weaker conditions.

Gaussian concentration bounds for eigenvalues of quasi-local observables are proven.

Abstract

We establish transportation cost inequalities (TCI) with respect to the quantum Wasserstein distance by introducing quantum extensions of well-known classical methods: first, using a non-commutative version of Ollivier's coarse Ricci curvature, we prove that high temperature Gibbs states of commuting Hamiltonians on arbitrary hypergraphs $H=(V,E)$ satisfy a TCI with constant scaling as $O(|V|)$. Second, we argue that the temperature range for which the TCI holds can be enlarged by relating it to recently established modified logarithmic Sobolev inequalities. Third, we prove that the inequality still holds for fixed points of arbitrary reversible local quantum Markov semigroups on regular lattices, albeit with slightly worsened constants, under a seemingly weaker condition of local indistinguishability of the fixed points. Finally, we use our framework to prove Gaussian concentration bounds for the distribution of eigenvalues of quasi-local observables and argue the usefulness of the TCI in proving the equivalence of the canonical and microcanonical ensembles and an exponential improvement over the weak Eigenstate Thermalization Hypothesis.