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December 2011 An asymptotic error bound for testing multiple quantum hypotheses
Michael Nussbaum, Arleta Szkoła
Ann. Statist. 39(6): 3211-3233 (December 2011). DOI: 10.1214/11-AOS933


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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Michael Nussbaum. Arleta Szkoła. "An asymptotic error bound for testing multiple quantum hypotheses." Ann. Statist. 39 (6) 3211 - 3233, December 2011.


Published: December 2011
First available in Project Euclid: 9 May 2012

zbMATH: 1246.62226
MathSciNet: MR3012406
Digital Object Identifier: 10.1214/11-AOS933

Primary: 62G10 , 62P35

Keywords: Bayesian discrimination , density operators , exponential error rate , Holevo–Helstrom tests , quantum Chernoff bound , quantum statistics

Rights: Copyright © 2011 Institute of Mathematical Statistics

Vol.39 • No. 6 • December 2011
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