Conclusive Discrimination by $N$ Sequential Receivers between $r\geq2$ Arbitrary Quantum States

Abstract

In the present paper, we develop a general mathematical framework for discrimination between $r\geq2$ quantum states by $N\geq1$ sequential receivers for the case in which every receiver obtains a conclusive result. This type of discrimination constitutes an $N$-sequential extension of the minimum-error discrimination by one receiver. The developed general framework, which is valid for a conclusive discrimination between any number $r\geq2$ of quantum states, pure or mixed, of an arbitrary dimension and any number $N\geq1$ of sequential receivers, is based on the notion of a quantum state instrument, and this allows us to derive new important general results. In particular, we find a general condition on $r\geq2$ quantum states under which, within the strategy in which all types of receivers’ quantum measurements are allowed, the optimal success probability of the $N$-sequential conclusive discrimination between these $r\geq2$ states is equal to that of the first receiver for any number $N\geq2$ of further sequential receivers and specify the corresponding optimal protocol. Furthermore, we extend our general framework to include an $N$-sequential conclusive discrimination between $r\geq2$ arbitrary quantum states under a noisy communication. As an example, we analyze analytically and numerically a two-sequential conclusive discrimination between two qubit states via depolarizing quantum channels. The derived new general results are important both from the theoretical point of view and for the development of a successful multipartite quantum communication via noisy quantum channels.