The Annals of Statistics

The Chernoff lower bound for symmetric quantum hypothesis testing

Michael Nussbaum and Arleta Szkoła

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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.

Article information

Ann. Statist., Volume 37, Number 2 (2009), 1040-1057.

First available in Project Euclid: 10 March 2009

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Zentralblatt MATH identifier

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. The Chernoff lower bound for symmetric quantum hypothesis testing. Ann. Statist. 37 (2009), no. 2, 1040--1057. doi:10.1214/08-AOS593.

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