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Exponential error rates in multiple state discrimination on a quantum spin chain

We consider decision problems on finite sets of hypotheses represented by pairwise different shift-invariant states on a quantum spin chain. The decision in favor of one of the hypotheses is based on outputs of generalized measurements performed on local states on blocks of finite size. We assume existence of the mean quantum Chernoff distances of any pair of states from the given set and refer to the minimum of them as the mean generalized quantum Chernoff distance. We establish that this minimum specifies an asymptotic bound on the exponential decay of the averaged probability of rejecting the true state in increasing block size, if the mean quantum Chernoff distance of any pair of the hypothetic states is achievable as an asymptotic error exponent in the corresponding binary problem. This assumption is in particular fulfiled by shift-invariant product states (i.i.d. states). Further, we provide a constructive proof for the existence of a sequence of quantum tests in increasing block size, which achieves an asymptotic error exponent which is equal to the mean generalized quantum Chernoff distance of the given set of states up to a factor, which depends on the set itself. It can be arbitrary close to 1 and is not less than $1/m$ for $m$ being the number of different pairs of states from the set considered.