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Journal of Physics A Mathematical and General

Upper bound on the success probability of separation among quantum states

Daowen Qiu

Published Jul 30, 2002• 25 citations
Publisher

Abstract

Quantum state separation is a more general operation for identifying states than unambiguous discrimination. In this paper, we derive an upper bound on the success probability of separation among n states with arbitrary a priori probabilities, extending some of the important results given in the literature. This conclusion generalizes that obtained by Chefles and Barnett for separating two states having equal a priori probabilities. Some of the known bounds on the success probabilities of unambiguous discrimination such as the Ivanovic–Dieks–Peres limit, the more general limit by Jaeger and Shimony, and an upper bound for the case of unambiguously discriminating n states, are special cases of our results. Notably, we also give implicitly a different method to derive the upper bound on the probability of successful unambiguous discrimination among n states. Finally, we apply our conclusion to quantum cloning and then derive some upper bounds on the success probabilities for several probabilistic cloning machines.

DOI
10.1088/0305-4470/35/32/312

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Quantum state separation is a more general operation for identifying states than unambiguous discrimination. In this paper, we derive an upper bound on the success probability of separation among n states with arbitrary a priori probabilities, extending some of the important results given in the literature. This conclusion generalizes that obtained by Chefles and Barnett for separating two states having equal a priori probabilities. Some of the known bounds on the success probabilities of unambiguous discrimination such as the Ivanovic–Dieks–Peres limit, the more general limit by Jaeger and Shimony, and an upper bound for the case of unambiguously discriminating n states, are special cases of our results. Notably, we also give implicitly a different method to derive the upper bound on the probability of successful unambiguous discrimination among n states. Finally, we apply our conclusion to quantum cloning and then derive some upper bounds on the success probabilities for several probabilistic cloning machines.
Unambiguous discrimination and exact cloning reduce the square-overlap between quantum states, exemplifying the more general type of procedure we term state separation . We obtain the maximum probability with which two equiprobable quantum states can be separated by an arbitrary degree, and find that the established bounds on the success probabilities for discrimination and cloning are special cases of this general bound. The latter also gives the maximum probability of successfully producing N exact copies of a quantum system whose state is chosen secretly from a known pair, given M initial realizations of the state, where N > M . We also discuss the relationship between this bound and that on unambiguous state discrimination.
Unambiguous discrimination and exact cloning reduce the square-overlap between quantum states, exemplifying the more general type of procedure we term state separation. We obtain the maximum probability with which two equiprobable quantum states can be separated by an arbitrary degree, and find that the established bounds on the success probabilities for discrimination and cloning are special cases of this general bound. The latter also gives the maximum probability of successfully producing N exact copies of a quantum system whose state is chosen secretly from a known pair, given M initial realizations of the state, where N > M. We also discuss the relationship between this bound and that on unambiguous state discrimination.
In this paper, we present a general upper bound on the success probability for unambiguous discrimination among arbitrary mmm mixed quantum states with given a priori probability. We further analyze how this upper bound can be achievable by presenting a sufficient condition related to it, and we compare this upper bound with three other upper bounds. Moreover, for the issue of the unambiguous identification of mmm unknown multicopy qudit pure states, we also get a bound.
For the optimal success probability under minimum-error discrimination between r≥2r\geq2r≥2 arbitrary quantum states prepared with any a priori probabilities, we find new general analytical lower and upper bounds and specify the relations between these new general bounds and the general bounds known in the literature. We also present the example where the new general analytical bounds, lower and upper, on the optimal success probability are tighter than most of the general analytical bounds known in the literature. The new upper bound on the optimal success probability explicitly generalizes to r>2r>2r>2 the form of the Helstrom bound. For r=2r=2r=2, each of our new bounds, lower and upper, reduces to the Helstrom bound.
Quantum state discrimination is a fundamental task that is meaningful in quantum information theory. In this manuscript, we consider an extended unambiguous discrimination of quantum resources. First, we present an upper bound of the success probability for an extended unambiguous discrimination task in the non-asymptotic and asymptotic scenarios. Next, we generalize the task from quantum states to quantum channels. We present an upper bound of the success probability for the task under the adaptive strategy. Furthermore, we show the bound can be computed efficiently. Finally, compared with the classical generalized unambiguous discrimination, we show the advantage of the quantum with the aid of a quantifier based on a set of semi-definite positive operators.