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

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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.

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  1. For the constraint used in [13] on receivers’ quantum measurements, see Section 2 of that paper.

  2. This is, for example, the case in [13], where the receivers’ quantum measurements are described by specific quantum instruments.

  3. One and the same POV measure may correspond to different quantum state instruments, see Section 2.

  4. See the representation (6).

  5. See Section 2.


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The study by E.R. Loubenets in Section 2, Section 3, and Section 4.1 of this work was supported by the Russian Science Foundation under grant No 19-11-00086 and performed at the Steklov Mathematical Institute of Russian Academy of Sciences. The study by E.R. Loubenets in Section 5 and Section 6 was performed at the National Research University Higher School of Economics. The study by Min Namkung in Section 4.2, Section 5 and Section 6 was performed until August 2021 at the National Research University Higher School of Economics and, from September 2021, at the Kyung Hee University under the support from the National Research Foundation of Korea (NRF) grant (NRF2020M3E4A1080088) funded by the Korea government (Ministry of Science and ICT).

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Loubenets, E.R., Namkung, M. Conclusive Discrimination by \(N\) Sequential Receivers between \(r\geq2\) Arbitrary Quantum States. Russ. J. Math. Phys. 30, 219–238 (2023).

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