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Quantum Measurements and Quantum Metrology

Ed. by Paternostro, Mauro

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2299-114X
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Adaptive estimation and discrimination of Holevo-Werner channels

Thomas P. W. Cope / Stefano Pirandola
Published Online: 2017-12-29 | DOI: https://doi.org/10.1515/qmetro-2017-0006

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).Google Scholar

  • [2] M. Hayashi, Quantum Information Theory: Mathematical Foundation (Springer-Verlag Berlin Heidelberg, 2017).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).CrossrefGoogle Scholar

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

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

  • [6] H. J. Kimble, Nature 453, 1023 (2008).Google Scholar

  • [7] S. Pirandola, and S. L. Braunstein, Nature 532, 169 (2016).Google Scholar

  • [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).Google Scholar

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

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

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

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

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

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

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

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

  • [18] E. Diamanti and A. Leverrier, Entropy 17, 6072 (2015).Google Scholar

  • [19] V. C. Usenko and R. Filip, Entropy 18, 20 (2016).CrossrefGoogle 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).Google Scholar

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

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

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

  • [24] P. Kok, S. L. Braunstein and J. P. Dowling, J. Op. B 6, 8 (2004).Google Scholar

  • [25] V. Giovannetti, S. Lloyd and L. Maccone, Science 306, 1330 (2004).Google Scholar

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

  • [27] V. Giovannetti, S. Lloyd, and L. Maccone, Nature Photonics 5, 222-229 (2011).CrossrefGoogle Scholar

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

  • [29] M. G. A. Paris, Int. J. Quant. Inf. 7, 125 (2009).CrossrefGoogle Scholar

  • [30] M. Tsang, R. Nair, and X. Lu, Phys. Rev. X 6, 031033 (2016).CrossrefGoogle Scholar

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

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

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

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

  • [35] S. M. Barnett and S. Croke, Advances in Optics and Photonics 1, 238 (2009).Google Scholar

  • [36] C. Invernizzi, M. G. A. Paris, and S. Pirandola, Phys. Rev. A 84, 022334 (2011).CrossrefGoogle Scholar

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

  • [38] G. Spedalieri and S. L. Braunstein, Phys. Rev. A 90, 052307 (2014).Google Scholar

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

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

  • [41] G. Spedalieri, C. Lupo, S. Mancini, S. L. Braunstein, and S. Pirandola, Phys. Rev. A 86, 012315 (2012).CrossrefGoogle Scholar

  • [42] C. Lupo, S. Pirandola, V. Giovannetti, and S.Mancini, Phys. Rev. A 87, 062310 (2013).Google Scholar

  • [43] R. Nair, Phys. Rev. A 84, 032312 (2011).CrossrefGoogle Scholar

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

  • [45] A. Bisio, M. Dall’Arno, and G. M. D’Ariano, Phys. Rev. A 84, 012310 (2011).CrossrefGoogle Scholar

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

  • [47] S. Lloyd, Science 321, 1463 (2008).Google Scholar

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

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

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

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

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

  • [54] A. Uhlmann, Rep. Math. Phys. 9, 273 (1976).CrossrefGoogle Scholar

  • [55] R. Jozsa, Journal of Modern Optics 41, 2315 (1994).CrossrefGoogle Scholar

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

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

  • [58] J. Calsamiglia, R. Munoz-Tapia, L. Masanes, A. Acin, and E. Bagan, Phys. Rev. A 77, 032311 (2008).Google Scholar

  • [59] S. Pirandola, and S. Lloyd, Phys. Rev. A 78, 012331 (2008).Google Scholar

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

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

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

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

  • [65] R. F. Werner, Phys. Rev. A, 40, 4277 (Oct 1989).CrossrefGoogle Scholar

  • [66] F. Barrett, Phys. Rev. A 65, 042302 (2002).Google Scholar

  • [67] D. Z. Djokovic, Entropy 18, 216 (2016).CrossrefGoogle Scholar

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

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

  • [70] K. G. H. Vollbrecht and R. F. Werner, Phys. Rev. A 64, 062307 (2001).Google Scholar

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

  • [72] V. Vedral, and M. B. Plenio, Phys. Rev. A 57, 1619 (1998).Google Scholar

  • [73] V. Vedral, Rev. Mod. Phys. 74, 197 (2002).CrossrefGoogle Scholar

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

  • [75] M. Horodecki and P. Horodecki, Phys. Rev. A 59, 4206 (1999).Google Scholar

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

  • [78] D. Leung and W. Matthews, IEEE Trans. Info. Theory 61, 4486 (2015).Google Scholar

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

  • [80] R. Laurenza and S. Pirandola, Phys. Rev. A 96, 032318 (2017).CrossrefGoogle Scholar

  • [81] R. Laurenza, S. L. Braunstein, and S. Pirandola, Finite-Resource Teleportation Stretching for Continuous-Variable Systems, arXiv:1706.06065 (2017).Google Scholar

  • [82] T. P. W. Cope, L. Hetzel, L. Banchi, and S. Pirandola, Phys. Rev. A 96, 022323 (2017).CrossrefGoogle Scholar

  • [83] R. Laurenza, C. Lupo, G. Spedalieri, S. L. Braunstein, and S. Pirandola, Channel Simulation in Quantum Metrology, arXiv:1712.06603 (2017).Google Scholar

  • [84] R. D. Gill and S. Massar, Phys. Rev. A 61, 042312 (2000).Google Scholar

About the article

Received: 2017-11-14

Accepted: 2017-12-08

Published Online: 2017-12-29

Published in Print: 2017-12-20


Citation Information: Quantum Measurements and Quantum Metrology, Volume 4, Issue 1, Pages 44–52, ISSN (Online) 2299-114X, DOI: https://doi.org/10.1515/qmetro-2017-0006.

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