Skip to content
BY-NC-ND 4.0 license Open Access Published by De Gruyter Open Access December 29, 2017

Adaptive estimation and discrimination of Holevo-Werner channels

  • Thomas P. W. Cope and Stefano Pirandola EMAIL logo

Abstract

The class of quantum states known as Werner states have several interesting properties, which often serve to illuminate unusual properties of quantum information. Closely related to these states are the Holevo- Werner channels whose Choi matrices are Werner states. Exploiting the fact that these channels are teleportation covariant, and therefore simulable by teleportation, we compute the ultimate precision in the adaptive estimation of their channel-defining parameter. Similarly, we bound the minimum error probability affecting the adaptive discrimination of any two of these channels. In this case, we prove an analytical formula for the quantum Chernoff bound which also has a direct counterpart for the class of depolarizing channels. Our work exploits previous methods established in [Pirandola and Lupo, PRL 118, 100502 (2017)] to set the metrological limits associated with this interesting class of quantum channels at any finite dimension.

References

[1] M. A. Nielsen, and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University Press, Cambridge, 2000).Search in Google Scholar

[2] M. Hayashi, Quantum Information Theory: Mathematical Foundation (Springer-Verlag Berlin Heidelberg, 2017).10.1007/978-3-662-49725-8Search in Google Scholar

[3] C. Weedbrook, S. Pirandola, R. García-Patrón, N. J. Cerf, T. C. Ralph, J. H. Shapiro, and S. Lloyd, Rev. Mod. Phys. 84, 621 (2012).Search in Google Scholar

[4] S. L. Braunstein and P. van Loock, Rev. Mod. Phys. 77, 513 (2005).Search in Google Scholar

[5] U. L. Andersen, J. S. Neergaard-Nielsen, P. van Loock, and A. Furusawa, Nature Phys. 11, 713 (2015).Search in Google Scholar

[6] H. J. Kimble, Nature 453, 1023 (2008).10.1038/nature07127Search in Google Scholar PubMed

[7] S. Pirandola, and S. L. Braunstein, Nature 532, 169 (2016).10.1038/532169aSearch in Google Scholar PubMed

[8] C. H. Bennett and G. Brassard. Quantum Cryptography: Public Key Distribution and Coin Tossing, Proceedings of IEEE International Conference on Computers, Systems and Signal Processing, 175 (1984).Search in Google Scholar

[9] A.K. Ekert, Phys. Rev. Lett. 67, 661 (1991).Search in Google Scholar

[10] N. Gisin, G. Ribordy, W. Tittel, and H. Zbinden, Rev. Mod. Phys. 74, 145 (2002).Search in Google Scholar

[11] F. Grosshans, G. Van Ache, J. Wenger, R. Brouri, N. J. Cerf, and P. Grangier, Nature 421, 238 (2003).10.1038/nature01289Search in Google Scholar PubMed

[12] C. Weedbrook, A. M. Lance, W. P. Bowen, T. Symul, T. C. Ralph, and P. K. Lam, Phys. Rev. Lett. 93, 170504 (2004).Search in Google Scholar

[13] R. Colbeck, QuantumAnd Relativistic Protocols For SecureMulti-Party Computation (PhD thesis, University of Cambridge, 2006).Search in Google Scholar

[14] S. Pirandola, S. Mancini, S. Lloyd, and S. L. Braunstein, Nat. Phys. 4, 726 (2008).Search in Google Scholar

[15] S. L. Braunstein and S. Pirandola, Phys. Rev. Lett. 108, 130502 (2012).Search in Google Scholar

[16] M.Curty, B. Qi, H.K. Lo, Phys. Rev. Lett. 108, 130503 (2012).Search in Google Scholar

[17] S. Pirandola, C. Ottaviani, G. Spedalieri, C. Weedbrook, S. L. Braunstein, S. Lloyd, T. Ghering, C.S. Jacobsen, and U. L. Andersen, Nat. Photon. 9, 397 (2015).Search in Google Scholar

[18] E. Diamanti and A. Leverrier, Entropy 17, 6072 (2015).10.3390/e17096072Search in Google Scholar

[19] V. C. Usenko and R. Filip, Entropy 18, 20 (2016).10.3390/e18010020Search in Google Scholar

[20] P. Shor, Polynomial-Time Algorithms for Prime Factorization and Discrete Logarithms on a Quantum Computer, Proceedings of the 35th Annual Symposium on Foundations of Computer Science, Santa Fe (1994).Search in Google Scholar

[21] S. Lloyd and S. L. Braunstein, Phys. Rev. Lett. 82, 1784 (1999).Search in Google Scholar

[22] T. D. Ladd, F. Jelezko, R. Laflamme,Y. Nakamura, C. Monroe and J. L. O’Brien, Nature 464, 45 (2010).10.1038/nature08812Search in Google Scholar PubMed

[23] S. L. Braunstein and C. M. Caves, Phys. Rev. Lett. 72, 3439 (1994).Search in Google Scholar

[24] P. Kok, S. L. Braunstein and J. P. Dowling, J. Op. B 6, 8 (2004).10.1088/1464-4266/6/8/029Search in Google Scholar

[25] V. Giovannetti, S. Lloyd and L. Maccone, Science 306, 1330 (2004).10.1126/science.1104149Search in Google Scholar PubMed

[26] H. M. Wiseman and G. J. Milburn, Quantum Measurement and Control (Cambridge University Press, 2010).10.1017/CBO9780511813948Search in Google Scholar

[27] V. Giovannetti, S. Lloyd, and L. Maccone, Nature Photonics 5, 222-229 (2011).10.1038/nphoton.2011.35Search in Google Scholar

[28] G. Toth, I. Apellaniz, J. Phys. A:Math. Theor. 47, 424006 (2014).Search in Google Scholar

[29] M. G. A. Paris, Int. J. Quant. Inf. 7, 125 (2009).Search in Google Scholar

[30] M. Tsang, R. Nair, and X. Lu, Phys. Rev. X 6, 031033 (2016).10.1103/PhysRevX.6.031033Search in Google Scholar

[31] C. Lupo and S. Pirandola, Phys. Rev. Lett. 117, 190802 (2016).Search in Google Scholar

[32] R. Nair, and M. Tsang, Phys. Rev. Lett. 117, 190801 (2016).Search in Google Scholar

[33] S. Pirandola, and C. Lupo, Phys. Rev. Lett. 118, 100502 (2017); ibid. 119, 129901 (2017).Search in Google Scholar

[34] A. Chefles, Contemp. Phys. 41, 401 (2000).Search in Google Scholar

[35] S. M. Barnett and S. Croke, Advances in Optics and Photonics 1, 238 (2009).10.1364/AOP.1.000238Search in Google Scholar

[36] C. Invernizzi, M. G. A. Paris, and S. Pirandola, Phys. Rev. A 84, 022334 (2011).10.1103/PhysRevA.84.022334Search in Google Scholar

[37] K. M. R. Audenaert, M. Nussbaum, A. Szkola, and F. Verstraete, Commun. Math. Phys. 279, 251 (2008).Search in Google Scholar

[38] G. Spedalieri and S. L. Braunstein, Phys. Rev. A 90, 052307 (2014).10.1103/PhysRevA.90.052307Search in Google Scholar

[39] S. Pirandola, Phys. Rev. Lett. 106, 090504 (2011).Search in Google Scholar

[40] S. Pirandola, C. Lupo, V. Giovannetti, S. Mancini, and S. L. Braunstein, New J. Phys. 13, 113012 (2011).Search in Google Scholar

[41] G. Spedalieri, C. Lupo, S. Mancini, S. L. Braunstein, and S. Pirandola, Phys. Rev. A 86, 012315 (2012).10.1103/PhysRevA.86.012315Search in Google Scholar

[42] C. Lupo, S. Pirandola, V. Giovannetti, and S.Mancini, Phys. Rev. A 87, 062310 (2013).10.1103/PhysRevA.87.062310Search in Google Scholar

[43] R. Nair, Phys. Rev. A 84, 032312 (2011).10.1103/PhysRevA.84.032312Search in Google Scholar

[44] O. Hirota, arXiv:1108.4163 (2011).Search in Google Scholar

[45] A. Bisio, M. Dall’Arno, and G. M. D’Ariano, Phys. Rev. A 84, 012310 (2011).10.1103/PhysRevA.84.042330Search in Google Scholar

[46] M. Dall’Arno et al., Phys. Rev. A 85, 012308 (2012).Search in Google Scholar

[47] S. Lloyd, Science 321, 1463 (2008).10.1126/science.1160627Search in Google Scholar PubMed

[48] S.-H. Tan et al., Phys. Rev. Lett. 101, 253601 (2008).Search in Google Scholar

[49] S. Barzanjeh et al., Phys. Rev. Lett. 114, 080503 (2015).Search in Google Scholar

[50] C. Weedbrook, S. Pirandola, J. Thompson, V. Vedral, and M. Gu, New J. Phys. 18, 043027 (2016).Search in Google Scholar

[51] E. D. Lopaeva, I. Ruo Berchera, I. P. Degiovanni, S. Olivares, G. Brida, and M. Genovese, Phys. Rev. Lett. 110, 153603 (2013).Search in Google Scholar

[52] Z. Zhang, S. Mouradian, F. N.C. Wong, and J. H. Shapiro, Phys. Rev. Lett. 114, 110506 (2015).Search in Google Scholar

[53] C.W. Helstrom, Quantum Detection and Estimation Theory (New York: Academic, 1976).Search in Google Scholar

[54] A. Uhlmann, Rep. Math. Phys. 9, 273 (1976).Search in Google Scholar

[55] R. Jozsa, Journal of Modern Optics 41, 2315 (1994).10.1080/09500349414552171Search in Google Scholar

[56] L. Banchi, S. L. Braunstein, and S. Pirandola, Phys. Rev. Lett. 115, 260501 (2015).Search in Google Scholar

[57] K. M. R. Audenaert et al., Phys. Rev. Lett. 98, 160501 (2007).Search in Google Scholar

[58] J. Calsamiglia, R. Munoz-Tapia, L. Masanes, A. Acin, and E. Bagan, Phys. Rev. A 77, 032311 (2008).10.1103/PhysRevA.77.032311Search in Google Scholar

[59] S. Pirandola, and S. Lloyd, Phys. Rev. A 78, 012331 (2008).10.1103/PhysRevA.78.012331Search in Google Scholar

[60] C. H. Bennett, G. Brassard, C. Crepeau, R. Jozsa, A. Peres, and W. K. Wootters, Phys. Rev. Lett. 70, 1895 (1993).Search in Google Scholar

[61] S. L. Braunstein, and H. J. Kimble, Phys. Rev. Lett. 80, 869 (1998).Search in Google Scholar

[62] S. Pirandola, J. Eisert, C. Weedbrook, A. Furusawa, and S. L. Braunstein, Nat. Photon. 9, 641 (2015).Search in Google Scholar

[63] S. Pirandola, R. Laurenza, C. Ottaviani and L. Banchi, Nat. Comm. 8, 15043 (2017). See also arXiv:1510.08863 (2015).Search in Google Scholar

[64] S. Pirandola, S. L. Braunstein, R. Laurenza, C. Ottaviani, T. P. W. Cope, G. Spedalieri, and L. Banchi, Theory of Channel Simulation and Bounds for Private Communication, arXiv:1711.09909v1 (2017).10.1088/2058-9565/aac394Search in Google Scholar

[65] R. F. Werner, Phys. Rev. A, 40, 4277 (Oct 1989).10.1103/PhysRevA.40.4277Search in Google Scholar PubMed

[66] F. Barrett, Phys. Rev. A 65, 042302 (2002).10.1103/PhysRevA.65.042302Search in Google Scholar

[67] D. Z. Djokovic, Entropy 18, 216 (2016).10.3390/e18060216Search in Google Scholar

[68] R. F. Werner, and A.S. Holevo, J. Mat. Phys. 43, 4353 (2002).Search in Google Scholar

[69] S. L. Braustein, and C. M. Caves, G. J. Milburn, Ann. Phys. 247, 135 (1996).Search in Google Scholar

[70] K. G. H. Vollbrecht and R. F. Werner, Phys. Rev. A 64, 062307 (2001).10.1103/PhysRevA.64.062307Search in Google Scholar

[71] V. Vedral, M. B. Plenio, M. A. Rippin, and P. L. Knight, Phys. Rev. Lett. 78, 2275 (1997).Search in Google Scholar

[72] V. Vedral, and M. B. Plenio, Phys. Rev. A 57, 1619 (1998).10.1103/PhysRevA.57.1619Search in Google Scholar

[73] V. Vedral, Rev. Mod. Phys. 74, 197 (2002).Search in Google Scholar

[74] M. Fannes, B. Haegeman,M. Mosonyi and D. Vanpeteghem, arXiv:quant-ph/0410195 (2004).Search in Google Scholar

[75] M. Horodecki and P. Horodecki, Phys. Rev. A 59, 4206 (1999).10.1103/PhysRevA.59.4206Search in Google Scholar

[76] A. Muller-Hermes, Transposition in Quantum Information Theory (Master’s thesis, Technical University of Munich, 2012).Search in Google Scholar

[77] M. M. Wolf, Notes on “Quantum Channels & Operations” (see page 35). Available at https://wwwm5.ma.tum.de/foswiki/pub/M5/Allgemeines/Michael-Wolf/QChannelLecture.pdf.Search in Google Scholar

[78] D. Leung and W. Matthews, IEEE Trans. Info. Theory 61, 4486 (2015).10.1109/TIT.2015.2439953Search in Google Scholar

[79] S. Pirandola, Capacities of Repeater-Assisted Quantum Communications, arXiv:1601.00966 (2016).Search in Google Scholar

[80] R. Laurenza and S. Pirandola, Phys. Rev. A 96, 032318 (2017).10.1103/PhysRevA.96.032318Search in Google Scholar

[81] R. Laurenza, S. L. Braunstein, and S. Pirandola, Finite-Resource Teleportation Stretching for Continuous-Variable Systems, arXiv:1706.06065 (2017).10.1038/s41598-018-33332-ySearch in Google Scholar PubMed PubMed Central

[82] T. P. W. Cope, L. Hetzel, L. Banchi, and S. Pirandola, Phys. Rev. A 96, 022323 (2017).10.1103/PhysRevA.96.022323Search in Google Scholar

[83] R. Laurenza, C. Lupo, G. Spedalieri, S. L. Braunstein, and S. Pirandola, Channel Simulation in Quantum Metrology, arXiv:1712.06603 (2017).10.1515/qmetro-2018-0001Search in Google Scholar

[84] R. D. Gill and S. Massar, Phys. Rev. A 61, 042312 (2000).10.1103/PhysRevA.61.042312Search in Google Scholar

Received: 2017-11-14
Accepted: 2017-12-8
Published Online: 2017-12-29
Published in Print: 2017-12-20

© 2018

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

Downloaded on 2.12.2023 from https://www.degruyter.com/document/doi/10.1515/qmetro-2017-0006/html
Scroll to top button