Abstract
In this work, we provide a complete analysis to minimum-error discrimination of mixed four qubit states with arbitrary prior probabilities. For the complete analysis, the most important work to do is to find the necessary and sufficient conditions for the existence of null measurement operator. From the geometric structure of qubit states, we obtain the analytic condition for deciding the existence of a null operator in minimum-error measurement for four mixed qubit states, which also gives the necessary and sufficient conditions for every optimal POVM to have nonzero elements. Using the condition, we completely analyze minimum-error discrimination of four mixed qubit states with arbitrary prior probabilities.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Data availability
All data generated or used during the study appear in the submitted article.
References
Chefles, A.: Quantum state discrimination. Contemp. Phys. 41, 401 (2000)
Barnett, S.M., Croke, S.: Quantum state discrimination. Adv. Opt. Photon. 1, 238 (2009)
Bergou, J.A.: Discrimination of quantum states. J. Mod. Opt. 57, 160 (2010)
Bae, J., Kwek, L.C.: Quantum state discrimination and its applications. J. Phys. A Math. Theor. 48, 083001 (2015)
Helstrom, C.W.: Quantum Detection and Estimation Theory. Academic Press, New York (1976)
Holevo, A.S.: Probabilistic and Statistical Aspects of Quantum Theory. North-Holland, Amsterdam (1979)
Yuen, H.P., Kennedy, R.S., Lax, M.: Optimum testing of multiple hypotheses in quantum detection theory. IEEE Trans. Inf. Theory 21, 125 (1975)
Barnett, S.M., Croke, S.: On the conditions for discrimination between quantum states with minimum error. J. Phys. A Math. Theor. 42, 062001 (2009)
Herzog, U.: Minimum-error discrimination between a pure and a mixed two-qubit state. J. Opt. B Quantum Semiclass. Opt. 6, S24 (2004)
Ban, M., Kurokawa, K., Momose, R., Hirota, O.: Optimum measurements for discrimination among symmetric quantum states and parameter estimation. Int. J. Theor. Phys. 36, 1269 (1997)
Eldar, Y.C., Forney, G.D.: On quantum detection and the square-root measurement. IEEE Trans. Inf. Theory 47, 858 (2001)
Chou, C.L., Hsu, L.Y.: Minimum-error discrimination between symmetric mixed quantum states. Phys. Rev. A 68, 042305 (2003)
Mochon, C.: Family of generalized “pretty good’’ measurements and the minimal-error pure-state discrimination problems for which they are optimal. Phys. Rev. A 73, 032328 (2006)
Samsonov, B.F.: Minimum error discrimination problem for pure qubit states. Phys. Rev. A 80, 052305 (2009)
Bae, J., Hwang, W.Y.: Minimum-error discrimination of qubit states: methods, solutions, and properties. Phys. Rev. A 87, 012334 (2013)
Bae, J.: Structure of minimum-error quantum state discrimination. New. J. Phys. 15, 073037 (2013)
Ha, D., Kwon, Y.: Complete analysis for three-qubit mixed-state discrimination. Phys. Rev. A 87, 062302 (2013)
Ha, D., Kwon, Y.: Discriminating \(N\)-qudit states using geometric structure. Phys. Rev. A 90, 022330 (2014)
Ha, D., Kwon, Y.: Quantum nonlocality without entanglement: explicit dependence on prior probabilities of nonorthogonal mirror-symmetric states. NPJ Quantum Inf. 7(1), 1–6 (2021)
Deconinck, M.E., Terhal, B.M.: Qubit state discrimination. Phys. Rev. A 81, 062304 (2010)
Eldar, Y.C., Megretski, A., Verghese, G.C.: Designing optimal quantum detectors via semidefinite programming. IEEE Trans. Inf. Theory 49, 1007 (2003)
Boyd, S., Vandenberghe, L.: Convex Optimization. Cambridge University Press, UK (2004)
Nakahira, K., Kato, K., Usuda, T.S.: Generalized quantum state discrimination problems. Phys. Rev. A 91, 052304 (2015)
Berkovitz, L.D.: Convexity and Optimization in \({\mathbb{R} }^{n}\). Wiley, New York (2002)
Davies, E.B.: Information and quantum measurement. IEEE Trans. Inf. Theory 24, 596 (1978)
Hunter, K.: Results in optimal discrimination. AIP Conf. Proc. 734, 83 (2004)
Acknowledgements
This work is supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (NRF2018R1D1A1B07049420 and NRF2022R1F1A1064459) and Institute of Information and Communications Technology Planning and Evaluation (IITP) grant funded by the Korean government (MSIT) (No. 2020001343, Artificial Intelligence Convergence Research Center (Hanyang University ERICA)
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Ha, D., Kwon, Y. Complete analysis to minimum-error discrimination of four mixed qubit states with arbitrary prior probabilities. Quantum Inf Process 22, 67 (2023). https://doi.org/10.1007/s11128-022-03814-0
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s11128-022-03814-0