Asymptotic Error Rates in Quantum Hypothesis Testing

Abstract

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.

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“…The archetypal problem in this setting is as follows: given the ability to create copies of an unknown quantum state ρ picked from a known set S of quantum states, identify ρ with minimal probability of error. Some authors use the term "quantum hypothesis testing" for this problem [55]; others reserve this term for the case |S| = 2, where precise results have been obtained relating the optimal error probability to the number of copies of ρ consumed, and trade-offs between different kinds of error have been determined [22]. See the surveys [27,55] for detailed reviews of quantum state discrimination.…”

Quantum probes for fractional Gaussian processes

“…We thus need to analyze the quantity P e,n = 1 2 1 − Tr 1 2 (|ρ ⊗n γ 2 − ρ ⊗n γ 1 )| . The evaluation of the trace distance for increasing n may be difficult and for this reason, one usually resort to the quantum Chernoff bound, which gives an upper bound to the probability of error [56,57,58,59,60,61] P e,n ≤ 1 2 Q n where…”

Quantum Chernoff bound as a measure of the efficiency of quantum cloning for mixed states

“…The task of the observer is to determine the minimal probability of error for identifying the quantum state by performing quantum operations on the k copies [22]. In the particular case of equiprobable states, the minimal error probability of discriminating them in a measurement performed on k independent copies is [27,28] …”