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Conclusive discrimination between arbitrary quantum states by N sequential receivers

Elena R. Loubenets, Min Namkung

Published Feb 9, 2021• 1 citation
OpenAlex

Abstract

In the present article, we develop a general framework for the description of an $N$-sequential state discrimination, where each of $N$ receivers always obtains a conclusive result. For this new state discrimination scenario, we derive two mutually equivalent general representations of the success probability and prove that if one of two states, pure or mixed, is prepared by a sender, then the optimal success probability is given by the Helstrom bound for any number $N$ of sequential receivers. Furthermore, we specify receivers' indirect measurements resulting in the optimal $N$-sequential conclusive state discrimination protocol. The developed framework is true for any number $N$ of sequential receivers, any number of arbitrary quantum states, pure or mixed, to be discriminated, and all types of receivers' quantum measurements. The new general results derived within the developed framework are important both from the theoretical point of view and for a successful multipartite quantum communication even in the presence of a quantum noise.

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In the present article, we develop a general framework for the description of an $N$-sequential state discrimination, where each of $N$ receivers always obtains a conclusive result. For this new state discrimination scenario, we derive two mutually equivalent general representations of the success probability and prove that if one of two states, pure or mixed, is prepared by a sender, then the optimal success probability is given by the Helstrom bound for any number $N$ of sequential receivers. Furthermore, we specify receivers' indirect measurements resulting in the optimal $N$-sequential conclusive state discrimination protocol. The developed framework is true for any number $N$ of sequential receivers, any number of arbitrary quantum states, pure or mixed, to be discriminated, and all types of receivers' quantum measurements. The new general results derived within the developed framework are important both from the theoretical point of view and for a successful multipartite quantum communication even in the presence of a quantum noise.

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In the present article, we develop a general framework for the description of an NNN-sequential state discrimination, where each of NNN receivers always obtains a conclusive result. For this new state discrimination scenario, we derive two mutually equivalent general representations of the success probability and prove that if one of two states, pure or mixed, is prepared by a sender, then the optimal success probability is given by the Helstrom bound for any number NNN of sequential receivers. Furthermore, we specify receivers' indirect measurements resulting in the optimal NNN-sequential conclusive state discrimination protocol. The developed framework is true for any number NNN of sequential receivers, any number of arbitrary quantum states, pure or mixed, to be discriminated, and all types of receivers' quantum measurements. The new general results derived within the developed framework are important both from the theoretical point of view and for a successful multipartite quantum communication even in the presence of a quantum noise.
NNN-sequential Conclusive State Discrimination in a General Case
In the present article, we develop a general framework for the description of an NNN-sequential state discrimination, where each of NNN receivers always obtains a conclusive result. For this new state discrimination scenario, we derive two mutually equivalent general representations of the success probability and prove that if one of two states, pure or mixed, is prepared by a sender, then the optimal success probability is given by the Helstrom bound for any number NNN of sequential receivers. Furthermore, we specify receivers' indirect measurements resulting in the optimal NNN-sequential conclusive state discrimination protocol. The developed framework is true for any number NNN of sequential receivers, any number of arbitrary quantum states, pure or mixed, to be discriminated, and all types of receivers' quantum measurements. The new general results derived within the developed framework are important both from the theoretical point of view and for a successful multipartite quantum communication even in the presence of a quantum noise.
Conclusive Discrimination by NNN Sequential Receivers between r≥2r\geq2r≥2 Arbitrary Quantum States
In the present paper, we develop a general mathematical framework for discrimination between r≥2r\geq2r≥2 quantum states by N≥1N\geq1N≥1 sequential receivers for the case in which every receiver obtains a conclusive result. This type of discrimination constitutes an NNN -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≥2r\geq2r≥2 of quantum states, pure or mixed, of an arbitrary dimension and any number N≥1N\geq1N≥1 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≥2r\geq2r≥2 quantum states under which, within the strategy in which all types of receivers’ quantum measurements are allowed, the optimal success probability of the NNN -sequential conclusive discrimination between these r≥2r\geq2r≥2 states is equal to that of the first receiver for any number N≥2N\geq2N≥2 of further sequential receivers and specify the corresponding optimal protocol. Furthermore, we extend our general framework to include an NNN -sequential conclusive discrimination between r≥2r\geq2r≥2 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.
Recently, sequential state discrimination, as a quantum-key distribution protocol, has been proposed for multiple receivers. A previous study [J. A. Bergou et al., Phys. Rev. Lett. 111, 100501 (2013)] showed that every receiver could successfully perform a sequential state discrimination of two pure states with identical prior probabilities. In this study, we extend the sequential state discrimination to mixed states with arbitrary prior probability. First, we analytically obtain the condition of the receiver's optimal measurement. In addition, we show that the optimal probability for every receiver to share the mixed state prepared by the sender is not zero. Furthermore, we compare the sequential state discrimination to the strategies of quantum reproducing and quantum broadcasting. We find that there are cases in which, unlike that of the pure state, the sequential state discrimination of mixed states shows a better performance than the other strategies.
Abstract A general scenario for an N -sequential conclusive state discrimination introduced recently in Loubenets and Namkung [arXiv:2102.04747] can provide a multipartite quantum communication realizable in the presence of a noise. In the present article, we propose a new experimental scheme for the implementation of a sequential conclusive discrimination between binary coherent states via indirect measurements within the Jaynes–Cummings interaction model. We find that if the mean photon number is less than 1.6, then, for our two-sequential state discrimination scheme, the optimal success probability is larger than the one presented in Fields, Bergou, and Varga (2020, IEEE Int. Conf. Quant. Comp. Eng. ). We also show that, if the mean photon number is almost equal to 1.2, then the optimal success probability nearly approaches the Helstrom bound.
Classical mixtures of quantum states often give rise to decoherence and are generally considered detrimental to quantum processing. However, in the framework of sequential measurement, such mixtures can be beneficial for state discrimination. We investigate the sequential discrimination of mixed states and compare the results with those of pure states under the condition of equal fidelity. It is found that the successful probability of the mixed-state protocol is superior to the pure one under the equal-fidelity condition. It is shown that the difference between the sequential discrimination of pure and mixed states is more reliable under the equal-fidelity condition than under single-shot discrimination, and this difference increases with the mixability of the initial mixed states. For scenarios in which classical communication is allowed, the optimal successful probability of pure-state discriminations is larger than that for mixed states on the contrary. We also show that the classical mixture of basic vectors from quantum decoherence has a subtle impact on the communication channel induced by the coincidence of the maximal mutual information and optimal successful probability of sequential discrimination for pure states.
Generalized contextuality is known to be required in optimal strategies for quantum state discrimination protocols. More recently, sequential discrimination tasks have been studied; n players attempt to determine in which state a qubit was prepared, in such a way that they all have a finite probability of success. We consider the extent to which contextuality plays a role in sequential versions of both unambiguous and minimum error discrimination. In the standard nonsequential case where n = 1, we use the COPE formalism to demonstrate that the presence of contextuality is guaranteed not only for the optimal measurement, but for a specific set of nonoptimal measurements as well. In the sequential case n > 1, we show that the presence of contextuality depends on which states are prepared, and on the protocol (unambiguous or minimum error) selected.