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.