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.
We consider decision problems on finite sets of hypotheses represented by pairwise different shift-invariant states on a quantum spin chain. The decision in favor of one of the hypotheses is based on outcomes of generalized measurements performed on local states on blocks of finite size. We assume the existence of the mean quantum Chernoff distances of any pair of states from the given set and refer to the minimum of them as the mean generalized quantum Chernoff distance. We establish that this minimum specifies an asymptotic bound on the exponential rate of decay of the averaged probability of rejecting the true state in increasing block size, if the mean quantum Chernoff distance of any pair of the hypothetic states is achievable as an asymptotic error exponent in the corresponding binary problem. This assumption is, in particular, fulfilled by shift-invariant product states (independent and identically distributed states). Further, we provide a constructive proof for the existence of a sequence of quantum tests in increasing block length with an error exponent which equals, up to a factor, the mean generalized quantum Chernoff distance. The factor depends on the configuration of the hypothetic states with respect to the binary quantum Chernoff distances. It can be arbitrary close to 1 and is never less than 1/m for m being the number of different pairs of states.
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.
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 on measurements performed on n of copies of a finite dimensional quantum system. It is known that the exponent for the rate of decrease of the averaged error probability cannot exceed the multiple quantum Chernoff bound (MQCB) defined as the worst case (smallest) quantum Chernoff distance between any possible pair of the r states. This error exponent is attainable for r pure states, but for the general case of mixed states only attainability up to a factor 1/3 is known. Here we show that the MQCB is attainable for mixed states if there is a pair which is closer in quantum Chernoff distance than 1/6 times the distance between all other pairs.
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.