Skip to main content

Thank you for visiting nature.com. You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in Internet Explorer). In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript.

  • Article
  • Published:

Exact solution for the quantum and private capacities of bosonic dephasing channels

Nature Photonics volume 17, pages 525–530 (2023)Cite this article

Subjects

Abstract

The capacities of noisy quantum channels capture the ultimate rates of information transmission across quantum communication lines, and the quantum capacity plays a key role in determining the overhead of fault-tolerant quantum computation platforms. Closed formulae for these capacities in bosonic systems were lacking for a key class of non-Gaussian channels, bosonic dephasing channels, which are used to model noise affecting superconducting circuits and fibre-optic communication channels. Here we provide an exact calculation of the quantum, private, two-way assisted quantum and secret-key-agreement capacities of all bosonic dephasing channels. We prove that they are equal to the relative entropy of the distribution underlying the channel with respect to the uniform distribution, solving a problem that was originally posed over a decade ago.

This is a preview of subscription content, access via your institution

Access options

Buy this article

  • Purchase on Springer Link
  • Instant access to full article PDF
Buy now

Prices may be subject to local taxes which are calculated during checkout

Fig. 1: A depiction of a quantum communication protocol that uses the channel \({{{\mathcal{N}}}}\) a total of n times to send a quantum system M reliably.
Fig. 2: An LOCC-assisted protocol that involves n uses of the quantum channel \({{{\mathcal{N}}}}\), assumed to connect two spatially separated laboratories belonging to Alice and Bob.
Fig. 3: The capacities of the BDCs associated with the wrapped normal distribution (pγ), the von Mises distribution (pλ) and the wrapped Cauchy distribution (pκ).

Similar content being viewed by others

Data availability

No data sets were generated during this study.

References

  1. Nielsen, M. A. & Chuang, I. L., Quantum Computation and Quantum Information (Cambridge Univ. Press, 2000).

  2. Shor, P. W., Algorithms for quantum computation: discrete logarithms and factoring. In Proc. 35th Annual Symposium on Foundations of Computer Science 124–134 (IEEE, 1994)

  3. Childs, A. M., Maslov, D., Nam, Y., Ross, N. J. & Su, Y. Toward the first quantum simulation with quantum speedup. Proc. Natl Acad. Sci. USA 115, 9456–9461 (2018).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  4. Grover, L. K. A fast quantum mechanical algorithm for database search. In Proc. 28th ACM Symposium on Theory of Computing (STOC) 212–219 (Association for Computing Machinery, 1996).

  5. Harrow, A. W., Hassidim, A. & Lloyd, S. Quantum algorithm for linear systems of equations. Phys. Rev. Lett. 103, 150502 (2009).

    Article  ADS  MathSciNet  Google Scholar 

  6. Gilyén, A., Su, Y., Low, G. H. & Wiebe, N. Quantum singular value transformation and beyond: exponential improvements for quantum matrix arithmetics. In Proc. 51st Annual ACM SIGACT Symposium on Theory of Computing 193–204 (Association for Computing Machinery, 2019).

  7. Xu, F., Ma, X., Zhang, Q., Lo, H.-K. & Pan, J.-W. Secure quantum key distribution with realistic devices. Rev. Mod. Phys. 92, 025002 (2020).

    Article  ADS  MathSciNet  Google Scholar 

  8. Schlosshauer, M. Decoherence, the measurement problem, and interpretations of quantum mechanics. Rev. Mod. Phys. 76, 1267–1305 (2005).

    Article  ADS  Google Scholar 

  9. Brito, F., DiVincenzo, D. P., Koch, R. H. & Steffen, M. Efficient one- and two-qubit pulsed gates for an oscillator-stabilized Josephson qubit. New J. Phys. 10, 033027 (2008).

    Article  ADS  Google Scholar 

  10. Taylor, J. M. et al. Fault-tolerant architecture for quantum computation using electrically controlled semiconductor spins. Nat. Phys. 1, 177–183 (2005).

    Article  Google Scholar 

  11. Ospelkaus, C. et al. Trapped-ion quantum logic gates based on oscillating magnetic fields. Phys. Rev. Lett. 101, 090502 (2008).

    Article  ADS  Google Scholar 

  12. Wanser, K. H. Fundamental phase noise limit in optical fibres due to temperature fluctuations. Electron. Lett. 28, 53–54(1) (1992).

    Article  Google Scholar 

  13. Gordon, J. P. & Mollenauer, L. F. Phase noise in photonic communications systems using linear amplifiers. Opt. Lett. 15, 1351–1353 (1990).

    Article  ADS  Google Scholar 

  14. Derickson, D. Fiber Optic Test and Measurement (Prentice Hall, 1998).

  15. Bartlett, S. D., Rudolph, T. & Spekkens, R. W. Reference frames, superselection rules, and quantum information. Rev. Mod. Phys. 79, 555–609 (2007).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  16. Jiang, L.-Z. & Chen, X.-Y. Evaluating the quantum capacity of bosonic dephasing channel. Proc. SPIE 7846, 784613 (2010).

    Article  Google Scholar 

  17. Arqand, A., Memarzadeh, L. & Mancini, S. Quantum capacity of a bosonic dephasing channel. Phys. Rev. A 102, 042413 (2020).

    Article  ADS  MathSciNet  Google Scholar 

  18. Devetak, I. & Shor, P. W. The capacity of a quantum channel for simultaneous transmission of classical and quantum information. Commun. Math. Phys. 256, 287–303 (2005).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  19. Hayashi, M. Quantum Information Theory: Mathematical Foundation 2nd edn (Springer, 2017).

  20. Wilde, M. M. Quantum Information Theory 2nd edn (Cambridge Univ. Press, 2017).

  21. Fawzi, O., Müller-Hermes, A. & Shayeghi, A. A lower bound on the space overhead of fault-tolerant quantum computation. In Proc. 13th Innovations in Theoretical Computer Science Conference (ITCS 2022) Vol. 215 (ed. Braverman, M.) 68:1–68:20 (Schloss Dagstuhl–Leibniz-Zentrum für Informatik, 2022).

  22. Bennett, C. H., DiVincenzo, D. P., Smolin, J. A. & Wootters, W. K. Mixed-state entanglement and quantum error correction. Phys. Rev. A 54, 3824–3851 (1996).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  23. Takeoka, M., Guha, S. & Wilde, M. M. The squashed entanglement of a quantum channel. IEEE Trans. Inf. Theory 60, 4987–4998 (2014).

    Article  MathSciNet  MATH  Google Scholar 

  24. Khatri, S. & Wilde, M. M. Principles of quantum communication theory: a modern approach. Preprint at https://arxiv.org/abs/2011.04672v1 (2020).

  25. Wilde, M. M., Tomamichel, M. & Berta, M. Converse bounds for private communication over quantum channels. IEEE Trans. Inf. Theory 63, 1792–1817 (2017).

    Article  MathSciNet  MATH  Google Scholar 

  26. Wehner, S., Elkouss, D. & Hanson, R. Quantum internet: A vision for the road ahead. Science 362, eaam9288 (2018).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  27. Arqand, A., Memarzadeh, L. & Mancini, S. Energy-constrained LOCC-assisted quantum capacity of bosonic dephasing channel. Preprint at https://arxiv.org/abs/2111.04173 (2021).

  28. Holevo, A. S. & Werner, R. F. Evaluating capacities of bosonic Gaussian channels. Phys. Rev. A 63, 032312 (2001).

    Article  ADS  Google Scholar 

  29. Wolf, M. M., Pérez-García, D. & Giedke, G. Quantum capacities of bosonic channels. Phys. Rev. Lett. 98, 130501 (2007).

    Article  ADS  Google Scholar 

  30. van Erven, T. & Harremos, P. Rényi divergence and Kullback–Leibler divergence. IEEE Trans. Inf. Theory 60, 3797–3820 (2014).

    Article  MATH  Google Scholar 

  31. Fanizza, M., Rosati, M., Skotiniotis, M., Calsamiglia, J. & Giovannetti, V. Squeezing-enhanced communication without a phase reference. Quantum 5, 608 (2021).

    Article  Google Scholar 

  32. Zhuang, Q. Quantum-enabled communication without a phase reference. Phys. Rev. Lett. 126, 060502 (2021).

    Article  ADS  MathSciNet  Google Scholar 

  33. Giovannetti, V. et al. Classical capacity of the lossy bosonic channel: the exact solution. Phys. Rev. Lett. 92, 027902 (2004).

    Article  ADS  Google Scholar 

  34. Giovannetti, V., García-Patrón, R., J. Cerf, N. & Holevo, A. S. Ultimate classical communication rates of quantum optical channels. Nat. Photonics 8, 796–800 (2014).

    Article  ADS  Google Scholar 

  35. Giovannetti, V., Lloyd, S., Maccone, L. & Shor, P. W. Entanglement assisted capacity of the broadband lossy channel. Phys. Rev. Lett. 91, 047901 (2003).

    Article  ADS  Google Scholar 

  36. Pirandola, S., Laurenza, R., Ottaviani, C. & Banchi, L. Fundamental limits of repeaterless quantum communications. Nat. Commun. 8, 15043 (2017).

    Article  ADS  Google Scholar 

  37. Shor, P. W. Scheme for reducing decoherence in quantum computer memory. Phys. Rev. A 52, R2493–R2496 (1995).

    Article  ADS  Google Scholar 

  38. Lidar, D. A. & Brun, T. A. (eds) Quantum Error Correction (Cambridge Univ. Press, 2013).

  39. DiVincenzo, D. P., Shor, P. W. & Smolin, J. A. Quantum-channel capacity of very noisy channels. Phys. Rev. A 57, 830–839 (1998).

    Article  ADS  Google Scholar 

  40. Smith, G. & Smolin, J. A. Degenerate quantum codes for Pauli channels. Phys. Rev. Lett. 98, 030501 (2007).

    Article  ADS  Google Scholar 

  41. Leditzky, F., Leung, D. & Smith, G. Dephrasure channel and superadditivity of coherent information. Phys. Rev. Lett. 121, 160501 (2018).

    Article  ADS  MathSciNet  Google Scholar 

  42. Devetak, I. & Winter, A. Distillation of secret key and entanglement from quantum states. Proc. R. Soc. A 461, 207–235 (2005).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  43. Devetak, I. The private classical capacity and quantum capacity of a quantum channel. IEEE Trans. Inf. Theory, 51, 44–55 (2005).

  44. Renes, J. M., Dupuis, F. & Renner, R. Efficient polar coding of quantum information. Phys. Rev. Lett. 109, 050504 (2012).

    Article  ADS  Google Scholar 

  45. Renes, J. M. & Wilde, M. M. Polar codes for private and quantum communication over arbitrary channels. IEEE Trans. Inf. Theory 60, 3090–3103 (2014).

    Article  MathSciNet  MATH  Google Scholar 

  46. Tomamichel, M., Wilde, M. M. & Winter, A. Strong converse rates for quantum communication. IEEE Trans. Inf. Theory 63, 715–727 (2017).

    Article  MathSciNet  MATH  Google Scholar 

  47. Szegő, G. Beiträge zur Theorie der Toeplitzschen Formen. Math. Z. 6, 167–202 (1920).

    Article  MathSciNet  MATH  Google Scholar 

  48. Serra-Capizzano, S. Test functions, growth conditions and Toeplitz matrices. Rend. Circ. Mat. Palermo Ser. II 68(Suppl.), 791–795 (2002).

    MathSciNet  MATH  Google Scholar 

  49. Bennett, C. H. et al. Teleporting an unknown quantum state via dual classical and Einstein–Podolsky–Rosen channels. Phys. Rev. Lett. 70, 1895–1899 (1993).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  50. Niset, J., Fiurasek, J. & Cerf, N. J. No-go theorem for Gaussian quantum error correction. Phys. Rev. Lett. 102, 120501 (2009).

    Article  ADS  Google Scholar 

  51. Müller-Hermes, A. Transposition in Quantum Information Theory. MSc thesis, Technische Universität München (2012).

  52. Wilde, M. M. & Qi, H. Energy-constrained private and quantum capacities of quantum channels. IEEE Trans. Inf. Theory 64, 7802–7827 (2018).

    Article  MathSciNet  MATH  Google Scholar 

  53. Bennett, C. H., DiVincenzo, D. P. & Smolin, J. A. Capacities of quantum erasure channels. Phys. Rev. Lett. 78, 3217–3220 (1997).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  54. Pirandola, S., García-Patrón, R., Braunstein, S. L. & Lloyd, S. Direct and reverse secret-key capacities of a quantum channel. Phys. Rev. Lett. 102, 050503 (2009).

    Article  ADS  MathSciNet  Google Scholar 

  55. Leviant, P., Xu, Q., Jiang, L. & Rosenblum, S. Quantum capacity and codes for the bosonic loss-dephasing channel. Quantum 6, 821 (2022).

    Article  Google Scholar 

  56. Sharma, K., Wilde, M. M., Adhikari, S. & Takeoka, M. Bounding the energy-constrained quantum and private capacities of phase-insensitive bosonic Gaussian channels. New J. Phys. 20, 063025 (2018).

    Article  ADS  Google Scholar 

  57. Rosati, M., Mari, A. & Giovannetti, V. Narrow bounds for the quantum capacity of thermal attenuators. Nat. Commun. 9, 4339 (2018).

    Article  ADS  Google Scholar 

  58. Noh, K., Albert, V. V. & Jiang, L. Quantum capacity bounds of Gaussian thermal loss channels and achievable rates with Gottesman–Kitaev–Preskill codes. IEEE Trans. Inf. Theory 65, 2563–2582 (2019).

    Article  MathSciNet  MATH  Google Scholar 

  59. Fanizza, M., Kianvash, F. & Giovannetti, V. Estimating quantum and private capacities of Gaussian channels via degradable extensions. Phys. Rev. Lett. 127, 210501 (2021).

    Article  ADS  MathSciNet  Google Scholar 

  60. Holevo, A. S. Entanglement-assisted capacities of constrained quantum channels. Theory Probab. Appl. 48, 243–255 (2004).

    Article  MathSciNet  Google Scholar 

  61. Horodecki, K., Horodecki, M., Horodecki, P. & Oppenheim, J. Secure key from bound entanglement. Phys. Rev. Lett. 94, 160502 (2005).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  62. Horodecki, K., Horodecki, M., Horodecki, P. & Oppenheim, J. General paradigm for distilling classical key from quantum states. IEEE Trans. Inf. Theory 55, 1898–1929 (2009).

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

We thank S. Mancini for discussions. L.L. was partially supported by the Alexander von Humboldt Foundation. M.M.W. acknowledges support from the National Science Foundation under grant no. 2014010.

Author information

Authors and Affiliations

  1. Institut für Theoretische Physik und IQST, Universität Ulm, Ulm, Germany

    Ludovico Lami

  2. QuSoft, Amsterdam, The Netherlands

    Ludovico Lami

  3. Korteweg–de Vries Institute for Mathematics, University of Amsterdam, Amsterdam, The Netherlands

    Ludovico Lami

  4. Institute for Theoretical Physics, University of Amsterdam, Amsterdam, The Netherlands

    Ludovico Lami

  5. Hearne Institute for Theoretical Physics, Department of Physics and Astronomy, and Center for Computation and Technology, Louisiana State University, Baton Rouge, LA, USA

    Mark M. Wilde

  6. School of Electrical and Computer Engineering, Cornell University, Ithaca, NY, USA

    Mark M. Wilde

Authors
  1. Ludovico Lami
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Mark M. Wilde
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

Both authors contributed to all aspects of this manuscript and to the writing of the paper.

Corresponding authors

Correspondence to Ludovico Lami or Mark M. Wilde.

Ethics declarations

Competing interests

The authors declare no competing interests.

Peer review

Peer review information

Nature Photonics thanks Javier Fonollosa and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

Additional information

Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Supplementary information

Supplementary Information

Supplementary Sections 1–4, Figs. 1–3, refs. 1–75 and detailed mathematical derivations of claims made in the text.

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.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Lami, L., Wilde, M.M. Exact solution for the quantum and private capacities of bosonic dephasing channels. Nat. Photon. 17, 525–530 (2023). https://doi.org/10.1038/s41566-023-01190-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1038/s41566-023-01190-4

This article is cited by

Search

Advanced search

Quick links

Nature Briefing

Sign up for the Nature Briefing newsletter — what matters in science, free to your inbox daily.

Get the most important science stories of the day, free in your inbox. Sign up for Nature Briefing