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

Abstract:We consider the problem of discriminating between two different states of a finite quantum system in the setting of large numbers of copies, and find a closed form expression for the asymptotic exponential rate at which the error probability tends to zero. This leads to the identification of the quantum generalisation of the classical Chernoff distance, which is the corresponding quantity in classical symmetric hypothesis testing.The proof relies on two new techniques introduced by the authors, which are also well suited to tackle the corresponding problem in asymmetric hypothesis testing, yielding the quantum generalisation of the classical Hoeffding bound. This has been done by Hayashi and Nagaoka for the special case where the states have full support.The goal of this paper is to present the proofs of these results in a unified way and in full generality, allowing hypothesis states with different supports. From the quantum Hoeffding bound, we then easily derive quantum Stein's Lemma and quantum Sanov's theorem. We give an in-depth treatment of the properties of the quantum Chernoff distance, and argue that it is a natural distance measure on the set of density operators, with a clear operational meaning.

Signal recovery in Gaussian white noise with variance tending to zero has served for some time as a representative model for nonparametric curve estimation, having all the essential traits in a pure form. The equivalence has mostly been stated informally, but an approximation in the sense of Le Cam's deficiency distance ∆ would make it precise. The models are then asymptotically equivalent for all purposes of statistical decision with bounded loss. In nonparametrics, a first result of this kind has recently been established for Gaussian regression (Brown and Low, 1993). We consider the analogous problem for the experiment given by n i. i. d. observations having density f on the unit interval. Our basic result concerns the parameter space of densities which are in a Hölder ball with exponent α > 1 2 and which are uniformly bounded away from zero. We show that an i. i. d. sample of size n with density f is globally asymptotically equivalent to a white noise experiment with drift f 1/2 and variance 1 4 n −1 . This represents a nonparametric analog of Le Cam's heteroscedastic Gaussian approximation in the finite dimensional case. The proof utilizes empirical process techniques related to the Hungarian construction. White noise models on f and log f are also considered, allowing for various "automatic" asymptotic risk bounds in the i. i. d. model from white noise. As first applications we discuss exact constants for L 2 and Hellinger loss.

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