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 investigate the ability of a quantum measurement device to discriminate two states or, generically, two hypotheses. In full generality, the measurement can be performed a number n of times, and arbitrary preprocessing of the states and postprocessing of the obtained data are allowed. There is an intrinsic error associated with the measurement device, which we aim to quantify, that limits its discrimination power. We minimize various error probabilities (averaged or constrained) over all pairs of n-partite input states. These probabilities, or their exponential rates of decrease in the case of large n, give measures of the discrimination power of the device. For the asymptotic rate of the averaged error probability, we obtain a Chernoff-type bound, dual to the standard Chernoff bound for which the state pair is fixed and the optimization is over all measurements. The key point in the derivation is that identical copies of input states become optimal in asymptotic settings. Optimal asymptotic rates are also obtained for constrained error probabilities, dual to Stein's lemma and Hoeffding's bound. We further show that adaptive protocols where the state preparer gets feedback from the measurer do not improve the asymptotic rates. These rates thus quantify the ultimate discrimination power of a measurement device.
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 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 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 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.