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.
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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{1236693159,
author = {Nussbaum, Michael and Szko\l a, Arleta},
title = {The Chernoff lower bound for symmetric quantum hypothesis testing},
journal = {Ann. Statist.},
volume = {37},
number = {1},
year = {2009},
pages = { 1040-1057},
language = {en},
url = {http://dml.mathdoc.fr/item/1236693159}
}
Nussbaum, Michael; Szkoła, Arleta. The Chernoff lower bound for symmetric quantum hypothesis testing. Ann. Statist., Tome 37 (2009) no. 1, pp. 1040-1057. http://gdmltest.u-ga.fr/item/1236693159/