Online identification of symmetric pure states

Gael Sentís, Esteban Martínez-Vargas, and Ramon Muñoz-Tapia

Física Teòrica: Informació i Fenòmens Quàntics, Departament de Física, Universitat Autònoma de Barcelona, 08193 Bellatera (Barcelona) Spain

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Abstract

We consider online strategies for discriminating between symmetric pure states with zero error when $n$ copies of the states are provided. Optimized online strategies involve local, possibly adaptive measurements on each copy and are optimal at each step, which makes them robust in front of particle losses or an abrupt termination of the discrimination process. We first review previous results on binary minimum and zero error discrimination with local measurements that achieve the maximum success probability set by optimizing over global measurements, highlighting their online features. We then extend these results to the case of zero error identification of three symmetric states with constant overlap. We provide optimal online schemes that attain global performance for any $n$ if the state overlaps are positive, and for odd $n$ if overlaps have a negative value. For arbitrary complex overlaps, we show compelling evidence that online schemes fail to reach optimal global performance. The online schemes that we describe only require to store the last outcome obtained in a classical memory, and adaptiveness of the measurements reduce to at most two changes, regardless of the value of $n$.

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► References

[1] A. Chefles, Quantum state discrimination, Contemporary Physics 41, 401 (2000), 10.1080/​00107510010002599.
https:/​/​doi.org/​10.1080/​00107510010002599

[2] S. M. Barnett and S. Croke, Quantum state discrimination, Advances in Optics and Photonics 1, 238 (2009), 10.1364/​AOP.1.000238.
https:/​/​doi.org/​10.1364/​AOP.1.000238

[3] J. A. Bergou, Discrimination of quantum states, Journal of Modern Optics 57, 160 (2010), 10.1080/​09500340903477756.
https:/​/​doi.org/​10.1080/​09500340903477756

[4] J. Bae and L.-C. Kwek, Quantum state discrimination and its applications, Journal of Physics A: Mathematical and Theoretical 48, 083001 (2015), 10.1088/​1751-8113/​48/​8/​083001.
https:/​/​doi.org/​10.1088/​1751-8113/​48/​8/​083001

[5] C. W. Helstrom, Quantum detection and estimation theory (Academic press) (1976).

[6] N. Gisin and R. Thew, Quantum communication, Nature Photonics 1, 165 (2007), 10.1038/​nphoton.2007.22.
https:/​/​doi.org/​10.1038/​nphoton.2007.22

[7] C. H. Bennett, Quantum cryptography using any two nonorthogonal states, Physical Review Letters 68, 3121 (1992), 10.1103/​PhysRevLett.68.3121.
https:/​/​doi.org/​10.1103/​PhysRevLett.68.3121

[8] N. Gisin, G. Ribordy, W. Tittel, and H. Zbinden, Quantum cryptography, Reviews of Modern Physics 74, 145 (2002), 10.1103/​RevModPhys.74.145.
https:/​/​doi.org/​10.1103/​RevModPhys.74.145

[9] A. Acín, J. Bae, E. Bagan, M. Baig, L. Masanes, and R. Muñoz-Tapia, Secrecy properties of quantum channels, Physical Review A 73, 012327 (2006), 10.1103/​PhysRevA.73.012327.
https:/​/​doi.org/​10.1103/​PhysRevA.73.012327

[10] R. Renner, Security of quantum key distribution, International Journal of Quantum Information 6, 1 (2008), 10.1142/​S0219749908003256.
https:/​/​doi.org/​10.1142/​S0219749908003256

[11] D. Bacon, A. Childs, and W. van Dam, From optimal measurement to efficient quantum algorithms for the hidden subgroup problem over semidirect product groups, in 46th Annual IEEE Symposium on Foundations of Computer Science (FOCS'05), vol. 2005, 469–478 (IEEE) (2005), ISBN 0-7695-2468-0, 10.1109/​SFCS.2005.38.
https:/​/​doi.org/​10.1109/​SFCS.2005.38

[12] J. Bae, W.-Y. Hwang, and Y.-D. Han, No-Signaling Principle Can Determine Optimal Quantum State Discrimination, Physical Review Letters 107, 170403 (2011), 10.1103/​PhysRevLett.107.170403.
https:/​/​doi.org/​10.1103/​PhysRevLett.107.170403

[13] R. Takagi and B. Regula, General Resource Theories in Quantum Mechanics and Beyond: Operational Characterization via Discrimination Tasks, Physical Review X 9, 031053 (2019), 10.1103/​PhysRevX.9.031053.
https:/​/​doi.org/​10.1103/​PhysRevX.9.031053

[14] M. Oszmaniec and T. Biswas, Operational relevance of resource theories of quantum measurements, Quantum 3, 133 (2019), 10.22331/​q-2019-04-26-133.
https:/​/​doi.org/​10.22331/​q-2019-04-26-133

[15] R. Uola, T. Kraft, J. Shang, X.-D. Yu, and O. Gühne, Quantifying Quantum Resources with Conic Programming, Physical Review Letters 122, 130404 (2019), 10.1103/​PhysRevLett.122.130404.
https:/​/​doi.org/​10.1103/​PhysRevLett.122.130404

[16] W. K. Wootters and W. H. Zurek, A single quantum cannot be cloned, Nature 299, 802 (1982), 10.1038/​299802a0.
https:/​/​doi.org/​10.1038/​299802a0

[17] J. Walgate, A. J. Short, L. Hardy, and V. Vedral, Local Distinguishability of Multipartite Orthogonal Quantum States, Physical Review Letters 85, 4972 (2000), 10.1103/​PhysRevLett.85.4972.
https:/​/​doi.org/​10.1103/​PhysRevLett.85.4972

[18] S. Virmani, M. Sacchi, M. Plenio, and D. Markham, Optimal local discrimination of two multipartite pure states, Physics Letters A 288, 62 (2001), 10.1016/​S0375-9601(01)00484-4.
https:/​/​doi.org/​10.1016/​S0375-9601(01)00484-4

[19] Y.-X. Chen and D. Yang, Optimal conclusive discrimination of two nonorthogonal pure product multipartite states through local operations, Physical Review A 64, 064303 (2001), 10.1103/​PhysRevA.64.064303.
https:/​/​doi.org/​10.1103/​PhysRevA.64.064303

[20] Y.-X. Chen and D. Yang, Optimally conclusive discrimination of nonorthogonal entangled states by local operations and classical communications, Physical Review A 65, 022320 (2002), 10.1103/​PhysRevA.65.022320.
https:/​/​doi.org/​10.1103/​PhysRevA.65.022320

[21] Z. Ji, H. Cao, and M. Ying, Optimal conclusive discrimination of two states can be achieved locally, Physical Review A 71, 032323 (2005), 10.1103/​PhysRevA.71.032323.
https:/​/​doi.org/​10.1103/​PhysRevA.71.032323

[22] A. Acín, E. Bagan, M. Baig, L. Masanes, and R. Muñoz-Tapia, Multiple-copy two-state discrimination with individual measurements, Physical Review A 71, 032338 (2005), 10.1103/​PhysRevA.71.032338.
https:/​/​doi.org/​10.1103/​PhysRevA.71.032338

[23] S. Croke, S. M. Barnett, and G. Weir, Optimal sequential measurements for bipartite state discrimination, Physical Review A 95, 052308 (2017), 10.1103/​PhysRevA.95.052308.
https:/​/​doi.org/​10.1103/​PhysRevA.95.052308

[24] A. Peres and W. K. Wootters, Optimal detection of quantum information, Physical Review Letters 66, 1119 (1991), 10.1103/​PhysRevLett.66.1119.
https:/​/​doi.org/​10.1103/​PhysRevLett.66.1119

[25] E. Chitambar and M.-H. Hsieh, Revisiting the optimal detection of quantum information, Physical Review A 88, 020302 (2013), 10.1103/​PhysRevA.88.020302.
https:/​/​doi.org/​10.1103/​PhysRevA.88.020302

[26] H.-C. Cheng, A. Winter, and N. Yu, Discrimination of quantum states under locality constraints in the many-copy setting, in 2021 IEEE International Symposium on Information Theory (ISIT), 1188–1193 (IEEE) (2021), 10.1109/​ISIT45174.2021.9518100.
https:/​/​doi.org/​10.1109/​ISIT45174.2021.9518100

[27] C. H. Bennett, D. P. DiVincenzo, C. A. Fuchs, T. Mor, E. Rains, P. W. Shor, J. A. Smolin, and W. K. Wootters, Quantum nonlocality without entanglement, Physical Review A 59, 1070 (1999), 10.1103/​PhysRevA.59.1070.
https:/​/​doi.org/​10.1103/​PhysRevA.59.1070

[28] C. A. Fuchs, Just two nonorthogonal quantum states, in P. Kumar, G. M. D'Ariano, and O. Hirota (eds.), Quantum Communication, Computing, and Measurement 2, 11–16 (Springer) (2002), 10.1007/​0-306-47097-7_2.
https:/​/​doi.org/​10.1007/​0-306-47097-7_2

[29] T. Eggeling and R. F. Werner, Hiding classical data in multipartite quantum states, Physical Review Letters 89, 097905 (2002), 10.1103/​PhysRevLett.89.097905.
https:/​/​doi.org/​10.1103/​PhysRevLett.89.097905

[30] S. S. Bhattacharya, S. Saha, T. Guha, and M. Banik, Nonlocality without entanglement: Quantum theory and beyond, Physical Review Research 2, 012068 (2020), 10.1103/​PhysRevResearch.2.012068.
https:/​/​doi.org/​10.1103/​PhysRevResearch.2.012068

[31] D. G. Fischer, S. H. Kienle, and M. Freyberger, Quantum-state estimation by self-learning measurements, Physical Review A 61, 032306 (2000), 10.1103/​PhysRevA.61.032306.
https:/​/​doi.org/​10.1103/​PhysRevA.61.032306

[32] A. Wald, Sequential Analysis, Dover books on advanced mathematics (Dover Publications) (1973), ISBN 9780486615790.

[33] E. Martínez Vargas, C. Hirche, G. Sentís, M. Skotiniotis, M. Carrizo, R. Muñoz-Tapia, and J. Calsamiglia, Quantum Sequential Hypothesis Testing, Physical Review Letters 126, 180502 (2021), 10.1103/​PhysRevLett.126.180502.
https:/​/​doi.org/​10.1103/​PhysRevLett.126.180502

[34] S. M. Barnett, Minimum-error discrimination between multiply symmetric states, Physical Review A 64, 030303 (2001), 10.1103/​PhysRevA.64.030303.
https:/​/​doi.org/​10.1103/​PhysRevA.64.030303

[35] J. A. Bergou, U. Futschik, and E. Feldman, Optimal unambiguous discrimination of pure quantum states, Physical Review Letters 108, 250502 (2012), 10.1103/​PhysRevLett.108.250502.
https:/​/​doi.org/​10.1103/​PhysRevLett.108.250502

[36] N. Dalla Pozza and G. Pierobon, Optimality of square-root measurements in quantum state discrimination, Physical Review A 91, 042334 (2015), 10.1103/​PhysRevA.91.042334.
https:/​/​doi.org/​10.1103/​PhysRevA.91.042334

[37] H. Krovi, S. Guha, Z. Dutton, and M. P. da Silva, Optimal measurements for symmetric quantum states with applications to optical communication, Physical Review A 92, 062333 (2015), 10.1103/​PhysRevA.92.062333.
https:/​/​doi.org/​10.1103/​PhysRevA.92.062333

[38] M. Skotiniotis, R. Hotz, J. Calsamiglia, and R. Muñoz Tapia, Identification of malfunctioning quantum devices, arXiv:1808.02729 (2018).
arXiv:1808.02729

[39] G. Sentís, J. Calsamiglia, and R. Muñoz Tapia, Exact identification of a quantum change point, Physical Review Letters 119, 140506 (2017), 10.1103/​PhysRevLett.119.140506.
https:/​/​doi.org/​10.1103/​PhysRevLett.119.140506

[40] E. Chitambar, R. Duan, and M.-H. Hsieh, When Do Local Operations and Classical Communication Suffice for Two-Qubit State Discrimination?, IEEE Transactions on Information Theory 60, 1549 (2014), 10.1109/​TIT.2013.2295356.
https:/​/​doi.org/​10.1109/​TIT.2013.2295356

[41] G. Sentís, E. Martínez-Vargas, and R. Muñoz Tapia, Online strategies for exactly identifying a quantum change point, Physical Review A 98, 052305 (2018), 10.1103/​PhysRevA.98.052305.
https:/​/​doi.org/​10.1103/​PhysRevA.98.052305

[42] K. Nakahira, K. Kato, and T. S. Usuda, Local unambiguous discrimination of symmetric ternary states, Physical Review A 99, 022316 (2019), 10.1103/​PhysRevA.99.022316.
https:/​/​doi.org/​10.1103/​PhysRevA.99.022316

[43] D. Brody and B. Meister, Minimum decision cost for quantum ensembles, Physical Review Letters 76, 1 (1996), 10.1103/​PhysRevLett.76.1.
https:/​/​doi.org/​10.1103/​PhysRevLett.76.1

[44] G. L. Nemhauser, Introduction to dynamic programming (John Wyley and Sons, New York) (1966).

[45] S. Brandsen, M. Lian, K. D. Stubbs, N. Rengaswamy, and H. D. Pfister, Adaptive procedures for discriminating between arbitrary tensor-product quantum states, in 2020 IEEE International Symposium on Information Theory (ISIT), 1933–1938 (IEEE) (2020), 10.1109/​ISIT44484.2020.9174234.
https:/​/​doi.org/​10.1109/​ISIT44484.2020.9174234

[46] S. Brandsen, K. D. Stubbs, and H. D. Pfister, Reinforcement learning with neural networks for quantum multiple hypothesis testing, in 2020 IEEE International Symposium on Information Theory (ISIT), 1897–1902 (IEEE) (2020), 10.1109/​ISIT44484.2020.9174150.
https:/​/​doi.org/​10.1109/​ISIT44484.2020.9174150

[47] K. Nakahira, K. Kato, and T. S. Usuda, Optimal discrimination of optical coherent states cannot always be realized by interfering with coherent light, photon counting, and feedback, Physical Review A 97, 022320 (2018), 10.1103/​PhysRevA.97.022320.
https:/​/​doi.org/​10.1103/​PhysRevA.97.022320

[48] A. Chefles, Unambiguous discrimination between linearly independent quantum states, Physics Letters A 239, 339 (1998), 10.1016/​S0375-9601(98)00064-4.
https:/​/​doi.org/​10.1016/​S0375-9601(98)00064-4

[49] A. Chefles, Unambiguous discrimination between linearly dependent states with multiple copies, Physical Review A 64, 062305 (2001), 10.1103/​PhysRevA.64.062305.
https:/​/​doi.org/​10.1103/​PhysRevA.64.062305

[50] Y. C. Eldar, A semidefinite programming approach to optimal unambiguous discrimination of quantum states, IEEE Transactions on Information Theory 49, 446 (2003), 10.1109/​TIT.2002.807291.
https:/​/​doi.org/​10.1109/​TIT.2002.807291

[51] E. Martínez Vargas and R. Muñoz Tapia, Certified answers for ordered quantum discrimination problems, Physical Review A 100, 042331 (2019), 10.1103/​PhysRevA.100.042331.
https:/​/​doi.org/​10.1103/​PhysRevA.100.042331

[52] E. Martínez Vargas, C. Hirche, G. Sentís, M. Skotiniotis, M. Carrizo, R. Muñoz Tapia, and J. Calsamiglia, Quantum sequential hypothesis testing, Physical Review Letters 126, 180502 (2021), 10.1103/​PhysRevLett.126.180502.
https:/​/​doi.org/​10.1103/​PhysRevLett.126.180502

[53] C. M. Caves, C. A. Fuchs, and R. Schack, Conditions for compatibility of quantum-state assignments, Physical Review A 66, 062111 (2002), 10.1103/​PhysRevA.66.062111.
https:/​/​doi.org/​10.1103/​PhysRevA.66.062111

[54] L.-M. Duan and G.-C. Guo, Probabilistic cloning and identification of linearly independent quantum states, Physical Review Letters 80, 4999 (1998), 10.1103/​PhysRevLett.80.4999.
https:/​/​doi.org/​10.1103/​PhysRevLett.80.4999

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