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We consider the problem of detecting the true quantum state among $r$possible ones, based of measurements performed on $n$ copies of afinite-dimensional quantum system. A special case is the problem ofdiscriminating between $r$ probability measures on a finite sample space, using$n$ i.i.d. observations. In this classical setting, it is known that theaveraged error probability decreases exponentially with exponent given by theworst case binary Chernoff bound between any possible pair of the $r$probability measures. Define analogously the multiple quantum Chernoff bound,considering all possible pairs of states. Recently, it has been shown that thisasymptotic error bound is attainable in the case of $r$ pure states, and thatit is unimprovable in general. Here we extend the attainability result to alarger class of $r$-tuples of states which are possibly mixed, but pairwiselinearly independent. We also construct a quantum detector which universallyattains the multiple quantum Chernoff bound up to a factor 1/3.

An asymptotic error bound for testing multiple quantum hypotheses