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.
We consider the problem of testing multiple quantum hypotheses
{ρ1⊗n,…,ρr⊗n}, where an arbitrary prior distribution is given and each of the
r hypotheses is
n copies of a quantum state. It is known that the minimal average error probability
Pe decays exponentially to zero, that is,
Pe=exp{−ξn+o(n)}. However, this error exponent
ξ is generally unknown, except for the case that
r=2. In this paper, we solve the long-standing open problem of identifying the above error exponent, by proving Nussbaum and Szkoła’s conjecture that
ξ=mini=jC(ρi,ρj). The right-hand side of this equality is called the multiple quantum Chernoff distance, and
C(ρi,ρj):=max0≤s≤1{−logTrρisρj1−s} has been previously identified as the optimal error exponent for testing two hypotheses,
ρi⊗n versus
ρj⊗n. The main ingredient of our proof is a new upper bound for the average error probability, for testing an ensemble of finite-dimensional, but otherwise general, quantum states. This upper bound, up to a states-dependent factor, matches the multiple-state generalization of Nussbaum and Szkoła’s lower bound. Specialized to the case
r=2, we give an alternative proof to the achievability of the binary-hypothesis Chernoff distance, which was originally proved by Audenaert et al.
Two types of errors can occur when discriminating pairs of quantum states. Asymmetric state discrimination involves minimizing the probability of one type of error, subject to a constraint on the other. We give explicit expressions bounding the set of achievable errors, using the trace norm, the fidelity, and the quantum Chernoff bound. The upper bound is asymptotically tight and the lower bound is exact for pure states. Unlike asymptotic bounds, our bounds give error values instead of exponents, so can give more precise results when applied to finite-copy state discrimination problems.
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 problem of detecting the true quantum state among r possible ones, based of measurements performed on n copies of a finite-dimensional quantum system. A special case is the problem of discriminating between r probability measures on a finite sample space, using n i.i.d. observations. In this classical setting, it is known that the averaged error probability decreases exponentially with exponent given by the worst case binary Chernoff bound between any possible pair of the r probability measures. Define analogously the multiple quantum Chernoff bound, considering all possible pairs of states. Recently, it has been shown that this asymptotic error bound is attainable in the case of r pure states, and that it is unimprovable in general. Here we extend the attainability result to a larger class of r-tuples of states which are possibly mixed, but pairwise linearly independent. We also construct a quantum detector which universally attains the multiple quantum Chernoff bound up to a factor 1/3.
The quantum Chernoff bound is a famous result about discriminating two different states in a setting where large numbers of copies are available, which gives the analytic asymptotic rate at which the minimal error probability decays to zero exponentially. It is however a challenging task to calculate the quantum Chernoff bound exactly in practical scenarios. In this paper, from the viewpoint of differential geometry, we demonstrate a remarkable link between the quantum Chernoff bound and Wigner-Yanase skew information. As a result, the quantum Chernoff bound can be estimated efficiently by virtue of the skew information. We present several examples to illustrate the efficiency of estimation about the quantum Chernoff bound via Wigner-Yanase skew information.