The Chernoff lower bound for symmetric quantum hypothesis testing

Details The Chernoff lower bound for symmetric quantum hypothesis testing The Chernoff lower bound for symmetric quantum hypothesis testing

2006-07-30

arxiv.org/abs/quant-ph/0607216v2

Annals of Statistics 2009, Vol. 37, No. 2, 1040-1057

Physics

Quantum Physics

Published in at http://dx.doi.org/10.1214/08-AOS593 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of

Scientific paper

10.1214/08-AOS593

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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