  We prove new concentration inequalities for quantum spin systems which apply
to any local observable measured on any product state or on any state with
exponentially decaying correlations. Our results do not require the spins to be
arranged in a regular lattice, and cover the case of observables that contain
terms acting on spins at arbitrary distance. Moreover, we introduce a local W1
distance, which quantifies the distinguishability of two states with respect to
local observables. We prove a transportation-cost inequality stating that the
local W1 distance between a generic state and a state with exponentially
decaying correlations is upper bounded by a function of their relative entropy.
Finally, we apply such inequality to prove the equivalence between the
canonical and microcanonical ensembles of quantum statistical mechanics and the
weak eigenstate thermalization hypothesis for the Hamiltonians whose Gibbs
states have exponentially decaying correlations.
