An asymptotic error bound for testing multiple quantum hypotheses
Michael Nussbaum and Arleta Szkola
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Submission date: 26. Aug. 2011 (revised version: November 2011)
published in: The annals of statistics, 39 (2011) 6, p. 3211-3233
DOI number (of the published article): 10.1214/11-AOS933
MSC-Numbers: 62P35, 62G10
Keywords and phrases: quantum statistics, Bayesian discrimination, quantum Chernoff bound, exponential error rate
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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 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 ex- ponent given by the worst 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 this asymptotic error bound is attainable in the case of r pure states, and that it is unimprovable in general. Here we extend the attainability result to a larger class of r-tuples of states which are possibly mixed, but pairwise linearly in- dependent. We also construct a quantum detector which universally attains the multiple quantum Chernoff bound up to a factor 1/3.