Quantum concentration inequalities

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.