Quantum Information Processing

, Volume 12, Issue 1, pp 641–683 | Cite as

Sequential, successive, and simultaneous decoders for entanglement-assisted classical communication

Article

Abstract

Bennett et al. showed that allowing shared entanglement between a sender and receiver before communication begins dramatically simplifies the theory of quantum channels, and these results suggest that it would be worthwhile to study other scenarios for entanglement-assisted classical communication. In this vein, the present paper makes several contributions to the theory of entanglement-assisted classical communication. First, we rephrase the Giovannetti–Lloyd–Maccone sequential decoding argument as a more general “packing lemma” and show that it gives an alternate way of achieving the entanglement-assisted classical capacity. Next, we show that a similar sequential decoder can achieve the Hsieh–Devetak–Winter region for entanglement-assisted classical communication over a multiple access channel. Third, we prove the existence of a quantum simultaneous decoder for entanglement-assisted classical communication over a multiple access channel with two senders. This result implies a solution of the quantum simultaneous decoding conjecture for unassisted classical communication over quantum multiple access channels with two senders, but the three-sender case still remains open (Sen recently and independently solved this unassisted two-sender case with a different technique). We then leverage this result to recover the known regions for unassisted and assisted quantum communication over a quantum multiple access channel, though our proof exploits a coherent quantum simultaneous decoder. Finally, we determine an achievable rate region for communication over an entanglement-assisted bosonic multiple access channel and compare it with the Yen-Shapiro outer bound for unassisted communication over the same channel.

Keywords

Quantum information theory Entanglement-assisted communication Quantum simultaneous decoding Quantum multiple access channel Bosonic channel 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Adami C., von Neumann N.J.C.: Capacity of noisy quantum channels. Phys. Rev. A 56(5), 3470–3483 (1997)ADSCrossRefGoogle Scholar
  2. 2.
    Bennett C.H., Brassard G., Crépeau C., Jozsa R., Peres A., Wootters W.K.: Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels. Phys. Rev. Lett. 70(13), 1895–1899 (1993)MathSciNetADSMATHCrossRefGoogle Scholar
  3. 3.
    Bennett C.H., Shor P.W., Smolin J.A., Thapliyal A.V.: Entanglement-assisted classical capacity of noisy quantum channels. Phys. Rev. Lett. 83(15), 3081–3084 (1999)ADSCrossRefGoogle Scholar
  4. 4.
    Bennett C.H., Shor P.W., Smolin J.A., Thapliyal A.V.: Entanglement-assisted capacity of a quantum channel and the reverse Shannon theorem. IEEE Trans. Inf. Theory 48, 2637–2655 (2002)MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Bennett C.H., Wiesner S.J.: Communication via one- and two-particle operators on Einstein-Podolsky-Rosen states. Phys. Rev. Lett. 69(20), 2881–2884 (1992)MathSciNetADSMATHCrossRefGoogle Scholar
  6. 6.
    Bowen G.: Quantum feedback channels. IEEE Trans. Inf. Theory 50(10), 2429–2434 (2004) arXiv: quant-ph/0209076MathSciNetCrossRefGoogle Scholar
  7. 7.
    Cover T.M., Thomas J.A.: Elements of Information Theory. Wiley-Interscience, New York (1991)MATHCrossRefGoogle Scholar
  8. 8.
    Csiszár I., Körner J.: Information Theory: Coding Theorems for Discrete Memoryless Systems, 2nd edn. Cambridge University Press, Cambridge (2011)MATHCrossRefGoogle Scholar
  9. 9.
    DattaN. , Renner R.: Smooth entropies and the quantum information spectrum. IEEE Trans. Inf. Theory 55(6), 2807–2815 (2009) arXiv:0801.0282CrossRefGoogle Scholar
  10. 10.
    Devetak I., Harrow A.W., Winter A.: A resource framework for quantum Shannon theory. IEEE Trans. Inf. Theory 54(10), 4587–4618 (2008)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Dutil, N.: Multiparty quantum protocols for assisted entanglement distillation. PhD thesis, McGill University (2011) arXiv:1105.4657Google Scholar
  12. 12.
    Eisert, J., Wolf, M.M.: Quantum Information with Continous Variables of Atoms and Light, chapter Gaussian quantum channels, pp. 23–42. Imperial College Press, London (2007). arXiv:quant-ph/0505151Google Scholar
  13. 13.
    El Gamal, A., Kim, Y.-H.: Lecture notes on network information theory (2010). arXiv:1001.3404Google Scholar
  14. 14.
    Fawzi, O., Hayden, P., Savov, I., Sen, P., Wilde, M.M.: Classical communication over a quantum interference channel (2011). arXiv:1102.2624Google Scholar
  15. 15.
    Feinstein A.: A new basic theorem of information theory. IEEE Trans. Inf. Theory 4(4), 2–22 (1954)MathSciNetGoogle Scholar
  16. 16.
    Gerry C., Knight P.: Introductory Quantum Optics. Cambridge University Press, Cambridge (2004)CrossRefGoogle Scholar
  17. 17.
    Giovannetti V., Guha S., Lloyd S., Maccone L., Shapiro J.H., Yuen H.P.: Classical capacity of the lossy bosonic channel: the exact solution. Phys. Rev. Lett. 92(2), 027902 (2004)ADSCrossRefGoogle Scholar
  18. 18.
    Giovannetti, V., Lloyd, S., Maccone, L.: Achieving the Holevo bound via sequential measurements (December 2010). arXiv:1012.0386Google Scholar
  19. 19.
    Giovannetti V., Lloyd S., Maccone L., Shor P.W.: Broadband channel capacities. Phys. Rev. A 68(6), 062323 (2003)ADSCrossRefGoogle Scholar
  20. 20.
    Giovannetti V., Lloyd S., Maccone L., Shor P.W.: Entanglement assisted capacity of the broadband lossy channel. Phys. Rev. Lett. 91(4), 047901 (2003)ADSCrossRefGoogle Scholar
  21. 21.
    Guha, S.: Multiple-User Quantum Information Theory for Optical Communication Channels. PhD thesis, Massachusetts Institute of Technology (2008)Google Scholar
  22. 22.
    Han T.S., Kobayashi K.: A new achievable rate region for the interference channel. IEEE Trans. Inf. Theory 27(1), 49–60 (1981)MathSciNetMATHCrossRefGoogle Scholar
  23. 23.
    Harrow A.W.: Coherent communication of classical messages. Phys. Rev. Lett. 92, 097902 (2004)MathSciNetADSCrossRefGoogle Scholar
  24. 24.
    Hayashi M., Nagaoka H.: General formulas for capacity of classical-quantum channels. IEEE Trans. Inf. Theory 49(7), 1753–1768 (2003)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Holevo A.S.: On entanglement assisted classical capacity. J. Math. Phys. 43(9), 4326–4333 (2002)MathSciNetADSMATHCrossRefGoogle Scholar
  26. 26.
    Holevo A.S., Werner R.F.: Evaluating capacities of bosonic Gaussian channels. Phys. Rev. A 63(3), 032312 (2001)ADSCrossRefGoogle Scholar
  27. 27.
    Horodecki M., Oppenheim J., Winter A.: Partial quantum information. Nature 436, 673–676 (2005)ADSCrossRefGoogle Scholar
  28. 28.
    Hsieh M.-H., Devetak I., Winter A.: Entanglement-assisted capacity of quantum multiple-access channels. IEEE Trans. Inf. Theory 54(7), 3078–3090 (2008)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Hsieh M.-H., Wilde M.M.: Entanglement-assisted communication of classical and quantum information. IEEE Trans. Inf. Theory 56(9), 4682–4704 (2010)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Hsieh M.-H., Wilde M.M.: Trading classical communication, quantum communication, and entanglement in quantum Shannon theory. IEEE Trans. Inf. Theory 56(9), 4705–4730 (2010)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Nielsen M.A., Chuang I.L.: Quantum Computation and Quantum Information. Cambridge University Press, Cambridge (2000)MATHGoogle Scholar
  32. 32.
    Ogawa T., Nagaoka H.: Making good codes for classical-quantum channel coding via quantum hypothesis testing. IEEE Trans. Inf. Theory 53(6), 2261–2266 (2007)MathSciNetCrossRefGoogle Scholar
  33. 33.
    Sen, P.: Private communication (2011)Google Scholar
  34. 34.
    Sen, P.: Sequential decoding for some channels with classical input and quantum output (2011)Google Scholar
  35. 35.
    Shannon C.E.: A mathematical theory of communication. Bell Syst. Tech. J. 27, 379–423 (1948)MathSciNetMATHGoogle Scholar
  36. 36.
    Weedbrook, C., Pirandola, S., Garcia-Patron, R., Cerf, N.J., Ralph, T.C., Shapiro, J.H., Lloyd, S.: Gaussian quantum information (2011 in preparation)Google Scholar
  37. 37.
    Wilde, M.M.: From Classical to Quantum Shannon Theory (2011). arXiv:1106.1445Google Scholar
  38. 38.
    Winter A.: Coding theorem and strong converse for quantum channels. IEEE Trans. Inf. Theory 45(7), 2481–2485 (1999)MATHCrossRefGoogle Scholar
  39. 39.
    Winter A.: The capacity of the quantum multiple access channel. IEEE Trans. Inf. Theory 47, 3059–3065 (2001)MATHCrossRefGoogle Scholar
  40. 40.
    Yard J., Hayden P., Devetak I.: Capacity theorems for quantum multiple-access channels: classical-quantum and quantum-quantum capacity regions. IEEE Trans. Inf. Theory 54(7), 3091–3113 (2008)MathSciNetCrossRefGoogle Scholar
  41. 41.
    Yen B.J., Shapiro J.H.: Multiple-access bosonic communications. Phys. Rev. A 72(6), 062312 (2005)ADSCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  1. 1.School of Computer ScienceMcGill UniversityMontrealCanada

Personalised recommendations