QSeminar: Giacomo de Palma

Speaker: Giacomo de Palma, University of Bologna

Title: The quantum Wasserstein distance of order 1

Abstract: We propose a generalization of the Wasserstein distance of order 1 to the quantum states of n qudits, and we prove a continuity bound for the von Neumann entropy with respect to the proposed distance. We also propose a generalization of the Lipschitz constant to quantum observables, and we prove a Gaussian concentration inequality for Lipschitz observables measured on the maximally mixed state. We then generalize the proposed quantum Wasserstein distance of order 1 to quantum spin systems on the lattice Z^d. We prove that local quantum commuting interactions above a critical temperature satisfy a transportation-cost inequality, which implies the uniqueness of their Gibbs states. Finally, we propose a modified definition of the quantum Lipschitz constant, and we prove a quasi-exponential concentration inequality for all the Lipschitz observables measured on any state with finite correlation length. Such inequality implies the equivalence of the canonical and microcanonical ensembles for all such states.