The Annals of Statistics

An asymptotic error bound for testing multiple quantum hypotheses

Michael Nussbaum and Arleta Szkoła

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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 exponent 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 independent. We also construct a quantum detector which universally attains the multiple quantum Chernoff bound up to a factor 1/3.

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Ann. Statist. Volume 39, Number 6 (2011), 3211-3233.

First available in Project Euclid: 9 May 2012

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Primary: 62P35: Applications to physics 62G10: Hypothesis testing

Quantum statistics density operators Bayesian discrimination exponential error rate Holevo–Helstrom tests quantum Chernoff bound


Nussbaum, Michael; Szkoła, Arleta. An asymptotic error bound for testing multiple quantum hypotheses. Ann. Statist. 39 (2011), no. 6, 3211--3233. doi:10.1214/11-AOS933.

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