Generalized quantum Chernoff bound

We establish a generalized quantum Chernoff bound for the discrimination of multiple sets of quantum states, thereby extending the classical and quantum Chernoff bounds to the general setting of composite and correlated quantum hypotheses. Specifically, we consider the task of distinguishing whether a quantum system is prepared in a state from one of several convex, compact sets of quantum states, each of which may exhibit arbitrary correlations. Assuming their stability under tensor product, we prove that the optimal error exponent for discrimination is precisely given by the regularized quantum Chernoff divergence between the sets. Furthermore, leveraging minimax theorems, we show that discriminating between sets of quantum states is no harder than discriminating between their worst-case elements in terms of error probability. This implies the existence of a universal optimal test that achieves the minimum error probability for all states in the sets, matching the performance of the optimal test for the most challenging states. We provide explicit characterizations of the universal optimal test in the binary composite case. Finally, we show that the maximum overlap between a pure state and a set of free states, a quantity that frequently arises in quantum resource theories, is equal to the quantum Chernoff divergence between the sets, thereby providing an operational interpretation of this quantity in the context of symmetric hypothesis testing.