  We consider the problem of whether the canonical and microcanonical ensembles
are locally equivalent for short-ranged quantum Hamiltonians of $N$ spins
arranged on a $d$-dimensional lattices. For any temperature for which the
system has a finite correlation length, we prove that the canonical and
microcanonical state are approximately equal on regions containing up to
$O(N^{1/(d+1)})$ spins. The proof rests on a variant of the Berry--Esseen
theorem for quantum lattice systems and ideas from quantum information theory.
