We consider the problem of discriminating two different quantum states in the setting of asymptotically many copies, and determine the optimal strategy that minimizes the total probability of error. This leads to the identification of the quantum Chernoff bound, thereby solving a long standing open problem. The bound reduces to the classical Chernoff bound when the quantum states under consideration commute. The quantum Chernoff bound is the natural symmetric distance measure between quantum states because of its clear operational meaning and because of the fact that it does not seem to share the undesirable features of other distance measures like the fidelity, the trace norm and the relative entropy.
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