Information transmission with continuous variable quantum erasure channels

Changchun Zhong, Changhun Oh, and Liang Jiang

Pritzker School of Molecular Engineering, University of Chicago, Chicago, IL 60637, USA

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Abstract

Quantum capacity, as the key figure of merit for a given quantum channel, upper bounds the channel's ability in transmitting quantum information. Identifying different types of channels, evaluating the corresponding quantum capacity, and finding the capacity-approaching coding scheme are the major tasks in quantum communication theory. Quantum channel in discrete variables has been discussed enormously based on various error models, while error model in the continuous variable channel has been less studied due to the infinite dimensional problem. In this paper, we investigate a general continuous variable quantum erasure channel. By defining an effective subspace of the continuous variable system, we find a continuous variable random coding model. We then derive the quantum capacity of the continuous variable erasure channel in the framework of decoupling theory. The discussion in this paper fills the gap of a quantum erasure channel in continuous variable setting and sheds light on the understanding of other types of continuous variable quantum channels.

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[1] M. Hayashi, S. Ishizaka, A. Kawachi, G. Kimura, and T. Ogawa, Introduction to quantum information science (Springer, 2014).
https:/​/​doi.org/​10.1007/​978-3-662-43502-1

[2] J. Watrous, The theory of quantum information (Cambridge university press, 2018).
https:/​/​doi.org/​10.1017/​9781316848142

[3] L. Gyongyosi, S. Imre, and H. V. Nguyen, A survey on quantum channel capacities, IEEE Communications Surveys & Tutorials 20, 1149 (2018).
https:/​/​doi.org/​10.1109/​COMST.2017.2786748

[4] C. H. Bennett and P. W. Shor, Quantum information theory, IEEE transactions on information theory 44, 2724 (1998).
https:/​/​doi.org/​10.1109/​18.720553

[5] P. Busch, P. Lahti, J.-P. Pellonpää, and K. Ylinen, Quantum measurement, Vol. 23 (Springer, 2016).
https:/​/​doi.org/​10.1007/​978-3-319-43389-9

[6] A. S. Holevo, The capacity of the quantum channel with general signal states, IEEE Transactions on Information Theory 44, 269 (1998).
https:/​/​doi.org/​10.1109/​18.651037

[7] H. Barnum, M. A. Nielsen, and B. Schumacher, Information transmission through a noisy quantum channel, Phys. Rev. A 57, 4153 (1998).
https:/​/​doi.org/​10.1103/​PhysRevA.57.4153

[8] S. Lloyd, Capacity of the noisy quantum channel, Phys. Rev. A 55, 1613 (1997).
https:/​/​doi.org/​10.1103/​PhysRevA.55.1613

[9] J. Eisert and M. M. Wolf, Gaussian quantum channels, arXiv preprint quant-ph/​0505151 (2005).
https:/​/​doi.org/​10.48550/​arXiv.quant-ph/​0505151
arXiv:quant-ph/0505151

[10] I. Devetak and P. W. Shor, The capacity of a quantum channel for simultaneous transmission of classical and quantum information, Communications in Mathematical Physics 256, 287 (2005).
https:/​/​doi.org/​10.1007/​s00220-005-1317-6

[11] A. S. Holevo, Quantum systems, channels, information, in Quantum Systems, Channels, Information (de Gruyter, 2019).
https:/​/​doi.org/​10.1515/​9783110273403

[12] M. Rosati, A. Mari, and V. Giovannetti, Narrow bounds for the quantum capacity of thermal attenuators, Nature communications 9, 1 (2018).
https:/​/​doi.org/​10.1038/​s41467-018-06848-0

[13] K. Sharma, M. M. Wilde, S. Adhikari, and M. Takeoka, Bounding the energy-constrained quantum and private capacities of phase-insensitive bosonic gaussian channels, New Journal of Physics 20, 063025 (2018).
https:/​/​doi.org/​10.1088/​1367-2630/​aac11a

[14] K. Jeong, Y. Lim, J. Kim, and S. Lee, New upper bounds on the quantum capacity for general attenuator and amplifier, in AIP Conference Proceedings, Vol. 2241 (AIP Publishing LLC, 2020) p. 020017.
https:/​/​doi.org/​10.1063/​5.0011402

[15] M. Grassl, T. Beth, and T. Pellizzari, Codes for the quantum erasure channel, Phys. Rev. A 56, 33 (1997).
https:/​/​doi.org/​10.1103/​PhysRevA.56.33

[16] C. H. Bennett, D. P. DiVincenzo, and J. A. Smolin, Capacities of quantum erasure channels, Phys. Rev. Lett. 78, 3217 (1997).
https:/​/​doi.org/​10.1103/​PhysRevLett.78.3217

[17] S. L. Braunstein and P. Van Loock, Quantum information with continuous variables, Reviews of modern physics 77, 513 (2005).
https:/​/​doi.org/​10.1103/​RevModPhys.77.513

[18] C. Weedbrook, S. Pirandola, R. García-Patrón, N. J. Cerf, T. C. Ralph, J. H. Shapiro, and S. Lloyd, Gaussian quantum information, Reviews of Modern Physics 84, 621 (2012).
https:/​/​doi.org/​10.1103/​RevModPhys.84.621

[19] D. Gottesman, A. Kitaev, and J. Preskill, Encoding a qubit in an oscillator, Physical Review A 64, 012310 (2001).
https:/​/​doi.org/​10.1103/​PhysRevA.64.012310

[20] W.-L. Ma, S. Puri, R. J. Schoelkopf, M. H. Devoret, S. Girvin, and L. Jiang, Quantum control of bosonic modes with superconducting circuits, Science Bulletin 66, 1789 (2021).
https:/​/​doi.org/​10.1016/​j.scib.2021.05.024

[21] J. Niset, U. L. Andersen, and N. J. Cerf, Experimentally feasible quantum erasure-correcting code for continuous variables, Phys. Rev. Lett. 101, 130503 (2008).
https:/​/​doi.org/​10.1103/​PhysRevLett.101.130503

[22] J. S. Sidhu, S. K. Joshi, M. Gündoğan, T. Brougham, D. Lowndes, L. Mazzarella, M. Krutzik, S. Mohapatra, D. Dequal, G. Vallone, et al., Advances in space quantum communications, IET Quantum Communication 2, 182 (2021).
https:/​/​doi.org/​10.1049/​qtc2.12015

[23] R. Klesse, Approximate quantum error correction, random codes, and quantum channel capacity, Phys. Rev. A 75, 062315 (2007).
https:/​/​doi.org/​10.1103/​PhysRevA.75.062315

[24] P. Hayden, M. Horodecki, A. Winter, and J. Yard, A decoupling approach to the quantum capacity, Open Systems & Information Dynamics 15, 7 (2008).
https:/​/​doi.org/​10.1142/​S1230161208000043

[25] P. Hayden and J. Preskill, Black holes as mirrors: quantum information in random subsystems, Journal of high energy physics 2007, 120 (2007).
https:/​/​doi.org/​10.1088/​1126-6708/​2007/​09/​120

[26] Q. Zhuang, T. Schuster, B. Yoshida, and N. Y. Yao, Scrambling and complexity in phase space, Phys. Rev. A 99, 062334 (2019).
https:/​/​doi.org/​10.1103/​PhysRevA.99.062334

[27] M. Fukuda and R. Koenig, Typical entanglement for gaussian states, Journal of Mathematical Physics 60, 112203 (2019).
https:/​/​doi.org/​10.1063/​1.5119950

[28] See the appendix for a brief review of the calculations for the discrete variable decoupling with any finite dimension.

[29] V. Paulsen, Completely bounded maps and operator algebras, 78 (Cambridge University Press, 2002).
https:/​/​doi.org/​10.1017/​CBO9780511546631

[30] B. Schumacher and M. A. Nielsen, Quantum data processing and error correction, Phys. Rev. A 54, 2629 (1996).
https:/​/​doi.org/​10.1103/​PhysRevA.54.2629

[31] B. Schumacher and M. D. Westmoreland, Approximate quantum error correction, Quantum Information Processing 1, 5 (2002).
https:/​/​doi.org/​10.1023/​A:1019653202562

[32] F. Dupuis, The decoupling approach to quantum information theory, arXiv preprint arXiv:1004.1641 (2010).
https:/​/​doi.org/​10.48550/​arXiv.1004.1641
arXiv:1004.1641

[33] M. Horodecki, J. Oppenheim, and A. Winter, Quantum state merging and negative information, Communications in Mathematical Physics 269, 107 (2007).
https:/​/​doi.org/​10.1007/​s00220-006-0118-x

[34] S. Choi, Y. Bao, X.-L. Qi, and E. Altman, Quantum error correction in scrambling dynamics and measurement-induced phase transition, Phys. Rev. Lett. 125, 030505 (2020).
https:/​/​doi.org/​10.1103/​PhysRevLett.125.030505

[35] B. Zhang and Q. Zhuang, Entanglement formation in continuous-variable random quantum networks, npj Quantum Information 7, 1 (2021).
https:/​/​doi.org/​10.1038/​s41534-021-00370-w

[36] A unitary design is a subset of the unitary group where the sample averages of certain polynomials over the set match that over the whole unitary group.

[37] C. E. Shannon, A mathematical theory of communication, The Bell system technical journal 27, 379 (1948).
https:/​/​doi.org/​10.1002/​j.1538-7305.1948.tb01338.x

[38] M. M. Wilde, Quantum information theory (Cambridge University Press, 2013).
https:/​/​doi.org/​10.1017/​9781316809976

[39] B. Collins and P. Śniady, Integration with respect to the haar measure on unitary, orthogonal and symplectic group, Communications in Mathematical Physics 264, 773 (2006).
https:/​/​doi.org/​10.1007/​s00220-006-1554-3

[40] V. V. Albert, K. Noh, K. Duivenvoorden, D. J. Young, R. T. Brierley, P. Reinhold, C. Vuillot, L. Li, C. Shen, S. M. Girvin, B. M. Terhal, and L. Jiang, Performance and structure of single-mode bosonic codes, Phys. Rev. A 97, 032346 (2018).
https:/​/​doi.org/​10.1103/​PhysRevA.97.032346

[41] K. Brádler and C. Adami, Black holes as bosonic gaussian channels, Phys. Rev. D 92, 025030 (2015).
https:/​/​doi.org/​10.1103/​PhysRevD.92.025030

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[2] Yanbo Lou, Yinghui Lv, Jiabin Wang, Shengshuai Liu, and Jietai Jing, "Orbital Angular Momentum Multiplexed Deterministic All-Optical Quantum Erasure-Correcting Code", Physical Review Letters 132 4, 040601 (2024).

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