Second Order Asymptotics for Quantum Hypothesis Testing

In the asymptotic theory of quantum hypothesis testing, the error probability of the first kind jumps sharply from zero to one when the error exponent of the second kind passes by the point of the relative entropy of the two states, in an increasing way. This is well known as the direct part and strong converse of quantum Stein’s lemma. Here we look into the behavior of this sudden change and have make it clear how the error of first kind grows according to a lower order of the error exponent of the second kind, and hence, we obtain the second order asymptotics for quantum hypothesis testing. Meanwhile, our analysis also yields tight bounds for the case of finite sample size. These results have nice applications in quantum information theory. Our method is elementary, based on basic linear algebra and probability theory. It deals with the achievability part and the converse part in a unified framework, with a clear geometric picture. Introduction. In the problem of asymptotic hypothesis testing, there are two hypotheses, each being many copies of independent and identically-distributed instances, occurring according to some given statistical description ρ or σ, respectively. Here the statistical descriptions are probability distributions in classical setting and quantum states in quantum mechanics. The task is to distinguish these two hypotheses with minimal error probabilities. Classically, this problem has been well understood [1]. Moving to the quantum case, it becomes much more difficult due to the non-commutativity of the quantum states ρ and σ, and the more complicated mechanics for observing the physical systems of interest(i.e., quantum measurement). Substantial achievements have already been made in the asymptotic theory of quantum hypothesis testing. Most notably, these include the establishment of the quantum Stein’s lemma with a strong converse [2, 3], the quantum Chernoff bound [4, 5], and the quantum Hoeffding bound [6–8]. Given a large number n of identical quantum systems, which are either of the state ρ⊗n (the null hypothesis) or of the state σ⊗n (the alternative hypothesis), we want to identify which state the systems belong to. This is achieved by doing a two outcome measurement {An, 1 −An}. We define two types of errors. Type I error (or the error of the first kind) is the probability that we falsely conclude that the state is σ while it is actually ρ, given by αn(An) := Tr ρ⊗n(1 −An); type II error (the error of the second kind) instead is the probability that we mistake σ for ρ, given by βn(An) := Trσ ⊗nAn. In an asymmetric situation, we want to minimize the type II error while only simply requiring that the type I error converges to 0. The quantum Stein’s lemma states that the maximal error exponent of type II is the relative entropy D(ρ∥σ): (direct part [2]) For arbitrary R < D(ρ∥σ), there exists a test {An, 1 −An} satisfying