## An asymptotic error bound for testing multiple quantum hypotheses.(English)Zbl 1246.62226

Summary: We consider the problem of detecting the true quantum state among $$r$$ possible ones, based of measurements performed on $$n$$ copies of a finite-dimensional quantum system. A special case is the problem of discriminating between $$r$$ probability measures on a finite sample space, using $$n$$ i.i.d. observations. In this classical setting, it is known that the averaged error probability decreases exponentially with the exponent given by the worst case binary Chernoff bound between any possible pair of the $$r$$ probability measures, and define analogously the multiple quantum Chernoff bound, considering all possible pairs of states. Recently, it has been shown that this asymptotic error bound is attainable in the case of $$r$$ pure states, and that it is unimprovable in general. We extend the attainability result to a larger class of $$r$$-tuples of states which are possibly mixed, but pairwise linearly independent. We also construct a quantum detector which universally attains the multiple quantum Chernoff bound up to a factor 1/3.

### MSC:

 62P35 Applications of statistics to physics 81P99 Foundations, quantum information and its processing, quantum axioms, and philosophy
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### References:

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