The asymptotic state discrimination problem with simple hypotheses is considered for a cubic lattice of bosons. A complete solution is provided for the problems of the Chernoff and the Hoeffding bounds and Stein’s lemma in the case when both hypotheses are Gaussian states with gauge- and translation-invariant quasifree parts.
The asymptotic discrimination problem of two quantum states is studied in the setting where measurements are required to be invariant under some symmetry group of the system. We consider various asymptotic error exponents in connection with the problems of the Chernoff bound, the Hoeffding bound, and Stein’s lemma, and derive bounds on these quantities in terms of their corresponding statistical distance measures. A special emphasis is put on the comparison of the performances of group-invariant and unrestricted measurements.
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.
In the problem of quantum state discrimination, one has to determine by measurements the state of a quantum system, based on the a priori side information that the true state is one of the two given and completely known states, ρ or σ. In general, it is not possible to decide the identity of the true state with certainty, and the optimal measurement strategy depends on whether the two possible errors (mistaking ρ for σ, or the other way around) are treated as of equal importance or not. Results on the quantum Chernoff and Hoeffding bounds and the quantum Stein's lemma show that, if several copies of the system are available then the optimal error probabilities decay exponentially in the number of copies, and the decay rate is given by a certain statistical distance between ρ and σ (the Chernoff distance, the Hoeffding distances, and the relative entropy, respectively). While these results provide a complete solution to the asymptotic problem, they are not completely satisfying from a practical point of view. Indeed, in realistic scenarios one has access only to finitely many copies of a system, and therefore it is desirable to have bounds on the error probabilities for finite sample size. In this paper we provide finite-size bounds on the so-called Stein errors, the Chernoff errors, the Hoeffding errors, and the mixed error probabilities related to the Chernoff and the Hoeffding errors.
We consider the asymmetric formulation of quantum hypothesis testing, where two quantum hypotheses have different associated costs. In this problem, the aim is to minimize the probability of false negatives and the optimal performance is provided by the quantum Hoeffding bound. After a brief review of these notions, we show how this bound can be simplified for pure states. We then provide a general recipe for its computation in the case of multimode Gaussian states, also showing its connection with other easier-to-compute lower bounds. In particular, we provide analytical formulas and numerical results for important classes of one- and two-mode Gaussian states.
A central yet mysterious problem in quantum information is the asymptotic distinguishability of quantum states under restricted measurements. Here we look at asymptotic asymmetric hypothesis testing for general multi-mode bosonic Gaussian states restricted to arbitrary collective Gaussian measurements. We establish a single-letter formula for the Stein exponent for many pairs of states, featuring two distinct contributions, corresponding to first and second moments, respectively. Remarkably, the Gaussian-measured relative entropy exhibits genuine non-additivity, but the only effect of additivity violations is to asymptotically decouple first and second moments. We prove this by establishing sufficient conditions for additivity of the second-moment term via novel log-determinant matrix inequalities. Our results not only identify a single-letter expression for the Gaussian Stein exponent in a wide range of practical situations, but also rule out any possible advantage from collective Gaussian measurements when considering centred Gaussian states.