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The Chernoff lower bound for symmetric quantum hypothesis testing

Michael Nussbaum·Arleta Szkoła
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Abstract

We consider symmetric hypothesis testing in quantum statistics, where the hypotheses are density operators on a finite-dimensional complex Hilbert space, representing states of a finite quantum system. We prove a lower bound on the asymptotic rate exponents of Bayesian error probabilities. The bound represents a quantum extension of the Chernoff bound, which gives the best asymptotically achievable error exponent in classical discrimination between two probability measures on a finite set. In our framework, the classical result is reproduced if the two hypothetic density operators commute. Recently, it has been shown elsewhere [Phys. Rev. Lett. 98 (2007) 160504] that the lower bound is achievable also in the generic quantum (noncommutative) case. This implies that our result is one part of the definitive quantum Chernoff bound.

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Michael Nussbaum, Arleta Szkoła. 2009-04-27. The Chernoff lower bound for symmetric quantum hypothesis testing. https://doi.org/10.1214/08-aos593

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