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

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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.

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Notes

  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.

References

  1. C. W. Helstrom, Quantum Detection and Estimation Theory, Academic Press, 1976.

    MATH  Google Scholar 

  2. A. S. Holevo, Probabilistic and Statistical Aspects of Quantum Theory, North-Holland, 1979.

    Google Scholar 

  3. J. A. Bergou, E. Feldman, and M. Hillery, “Extracting Information from a Qubit by Multiple Observers: Toward a Theory of Sequential State Discrimination”, Phys. Rev. Lett., 111 (2013), 100501.

    Article  ADS  Google Scholar 

  4. C.-Q. Pang, F.-L Zhang, L.-F. Xu, M.-L. Liang, and J.-L. Chen, “Sequential State Discrimination and Requirement of Quantum Dissonance”, Phys. Rev. A, 88 (2013), 052331.

    Article  ADS  Google Scholar 

  5. J.-H. Zhang, F.-L. Zhang, and M.-L. Liang, “Sequential State Discrimination with Quantum Correlation”, Quantum. Inf. Proces., 17 (2018), 260.

    Article  ADS  MathSciNet  MATH  Google Scholar 

  6. M. Namkung and Y. Kwon, “Optimal Sequential State Discrimination between Two Mixed Quantum States”, Phys. Rev. A, 96 (2017), 022318.

    Article  ADS  Google Scholar 

  7. M. Namkung and Y. Kwon, “Analysis of Optimal Sequential State Discrimination for Linearly Independent Pure Quantum States”, Sci. Rep., 8 (2018), 6515.

    Article  ADS  Google Scholar 

  8. M. Hillery and J. Mimih, “Sequential Discrimination of Qudits by Multiple Receivers”, J. Phys. A: Math. Theor., 50 (2017), 435301.

    Article  ADS  MATH  Google Scholar 

  9. M. Namkung and Y. Kwon, “Generalized Sequential State Discrimination for Multiparty QKD and Its Optical Implementation”, Sci. Rep., 10 (2020), 8247.

    Article  ADS  Google Scholar 

  10. M. A. Solis-Prosser, P. Gonzalez, J. Fuenzalida, S. Gomez, G. B. Xavier, A. Delgado, and G. Lima, “Experimental Multiparty Sequential State Discrimination”, Phys. Rev. A, 94 (2016), 042309.

    Article  ADS  Google Scholar 

  11. M. Namkung and Y. Kwon, “Sequential State Discrimination of Coherent States”, Sci. Rep., 8 (2018), 16915.

    Article  ADS  Google Scholar 

  12. T. Rudolph, R. W. Spekkens, and P. S. Turner, “Unambiguous Discrimination of Mixed States”, Phys. Rev. A, 68 (2003), 010301(R).

    Article  ADS  MathSciNet  Google Scholar 

  13. D. Fields, A. Varga, and J. A. Bergou, Sequential Measurements on Qubits by Multiple Observers: Joint Best Guess Strategy, IEEE International Conference on Quantum Computing and Engineering (QCE), 2020.

    Google Scholar 

  14. E. B. Davies, Quantum Theory of Open Systems, Academic Press, 1976.

    MATH  Google Scholar 

  15. P. Busch, M. Grabowski, and P. J. Lahti, Operational Quantum Physics, Springer, 1995.

    Book  MATH  Google Scholar 

  16. A. S. Holevo, Statistical Structure of Quantum Theory, Springer, Berlin, 2001.

    Book  MATH  Google Scholar 

  17. E. R. Loubenets, “Quantum Stochastic Approach to the Description of Quantum Measurements”, J. Phys. A: Math. Gen., 34 (2001), 7639–7675.

    Article  ADS  MathSciNet  MATH  Google Scholar 

  18. O. E. Barndoff-Nielson and E. R. Loubenets, “General Framework for the Behaviour of Continuously Observed Open Quantum Systems”, J. Phys. A, 35 (2002), 565–588.

    Article  ADS  MathSciNet  MATH  Google Scholar 

  19. M. E. Shirokov, “Entropy Reduction of Quantum Measurements”, J. Math. Phys., 52 (2011), 052202.

    Article  ADS  MathSciNet  MATH  Google Scholar 

  20. M. E. Shirokov, “On Properties of the Space of Quantum States and Their Application to the Construction of Entanglement Monotones”, Izv. Math., 74:4 (2010), 849–882.

    Article  MathSciNet  MATH  Google Scholar 

  21. K. Kraus, States, Effects, and Operations: Fundamental Notions of Quantum Theory, Springer, 1983.

    Book  MATH  Google Scholar 

  22. M. Ozawa, “Quantum Measuring Processes of Continuous Observables”, J. Math. Phys., 25 (1984), 79.

    Article  ADS  MathSciNet  Google Scholar 

  23. E. R. Loubenets, “General Lower and Upper Bounds under Minimum-Error Quantum State Discrimination”, Phys. Rev. A., 105 (2022), 032410.

    Article  ADS  MathSciNet  Google Scholar 

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Funding

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). https://doi.org/10.1134/S1061920823020085

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