Quantum Information Processing

, Volume 11, Issue 6, pp 1465–1501 | Cite as

Public and private resource trade-offs for a quantum channel

Article

Abstract

Collins and Popescu realized a powerful analogy between several resources in classical and quantum information theory. The Collins–Popescu analogy states that public classical communication, private classical communication, and secret key interact with one another somewhat similarly to the way that classical communication, quantum communication, and entanglement interact. This paper discusses the information-theoretic treatment of this analogy for the case of noisy quantum channels. We determine a capacity region for a quantum channel interacting with the noiseless resources of public classical communication, private classical communication, and secret key. We then compare this region with the classical-quantum-entanglement region from our prior efforts and explicitly observe the information-theoretic consequences of the strong correlations in entanglement and the lack of a super-dense coding protocol in the public-private-secret-key setting. The region simplifies for several realistic, physically-motivated channels such as entanglement-breaking channels, Hadamard channels, and quantum erasure channels, and we are able to compute and plot the region for several examples of these channels.

Keywords

Quantum Shannon theory Public classical communication Private classical communication Secret key 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Ahlswede R., Csiszár I.: Common randomness in information theory and cryptography—Part I: secret sharing. IEEE Trans. Inf. Theory 39, 1121–1132 (1993)MATHCrossRefGoogle Scholar
  2. 2.
    Ahlswede R., Csiszár I.: Common randomness in information theory and cryptography—Part II: CR-capacity. IEEE Trans. Inf. Theory 44, 225–240 (1998)MATHCrossRefGoogle Scholar
  3. 3.
    Bennett, C.H., Brassard, G.: Quantum cryptography: public key distribution and coin tossing. In: IEEE Proceedings of IEEE International Conference on Computers, Systems, and Signal Processing, pp. 175–179 (1984)Google Scholar
  4. 4.
    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, 1895–1899 (1993)MathSciNetADSMATHCrossRefGoogle Scholar
  5. 5.
    Bennett C.H., DiVincenzo D.P., Smolin J.A.: Capacities of quantum erasure channels. Phys. Rev. Lett. 78(16), 3217–3220 (1997). doi:10.1103/PhysRevLett.78.3217 MathSciNetADSMATHCrossRefGoogle Scholar
  6. 6.
    Bennett C.H., Wiesner S.J.: Communication via one-and two-particle operators on Einstein–Podolsky–Rosen states. Phys. Rev. Lett. 69, 2881–2884 (1992)MathSciNetADSMATHCrossRefGoogle Scholar
  7. 7.
    Boyd, S., Vandenberghe, L.: Convex Optimization. Cambridge University Press, Cambridge (2004). url:http://www.stanford.edu/~boyd/cvxbook/
  8. 8.
    Brádler K.: An infinite sequence of additive channels: the classical capacity of cloning channels. IEEE Trans. Inf. Theory. 57, 5497–5503 (2011)CrossRefGoogle Scholar
  9. 9.
    Brádler K., Dutil N., Hayden P., Muhammad A.: Conjugate degradability and the quantum capacity of cloning channels. J. Math. Phys. 51, 072201 (2010). doi:10.1063/1.3449555 MathSciNetADSCrossRefGoogle Scholar
  10. 10.
    Brádler, K., Hayden, P., Panangaden, P.: Private information via the Unruh effect. J. High Energy Phys. 2009(08), 074 (2009). url:http://stacks.iop.org/1126-6708/2009/i=08/a=074
  11. 11.
    Brádler K., Hayden P., Touchette D., Wilde M.M.: Trade-off capacities of the quantum Hadamard channels. Phys. Rev. A 81(6), 062312 (2010)ADSCrossRefGoogle Scholar
  12. 12.
    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(3), 033,027 (2008). url:http://stacks.iop.org/1367-2630/10/033027
  13. 13.
    Cai N., Winter A., Yeung R.W.: Quantum privacy and quantum wiretap channels. Prob. Inf. Transm. 40(4), 318–336 (2004). doi:10.1007/s11122-005-0002-x MathSciNetMATHGoogle Scholar
  14. 14.
    Collins D., Popescu S.: Classical analog of entanglement. Phys. Rev. A 65(3), 032,321 (2002). doi:10.1103/PhysRevA.65.032321 MathSciNetCrossRefGoogle Scholar
  15. 15.
    Csiszár I., Körner J.: Broadcast channels with confidential messages. IEEE Trans. Inf. Theory 24(3), 339–348 (1967)CrossRefGoogle Scholar
  16. 16.
    Devetak I.: The private classical capacity and quantum capacity of a quantum channel. IEEE Trans. Inf. Theory 51(1), 44–55 (2005)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Devetak I., Shor P.W.: The capacity of a quantum channel for simultaneous transmission of classical and quantum information. Commun. Math. Phys. 256(2), 287–303 (2005)MathSciNetADSMATHCrossRefGoogle Scholar
  18. 18.
    Devetak I., Winter A.: Relating quantum privacy and quantum coherence: an operational approach. Phys. Rev. Lett. 93, 080,501 (2004)MathSciNetGoogle Scholar
  19. 19.
    Devetak I., Winter A.: Distillation of secret key and entanglement from quantum states. Proc. Roy. Soc. A 461, 207–235 (2005)MathSciNetADSMATHCrossRefGoogle Scholar
  20. 20.
    Duan, R., Grassl, M., Ji, Z., Zeng, B.: Multi-error-correcting amplitude damping codes. In: Proceedings of the International Symposium on Information Theory. Austin, Texas, USA (2010). ArXiv:1001.2356Google Scholar
  21. 21.
    Gingrich R.M., Kok P., Lee H., Vatan F., Dowling J.P.: All linear optical quantum memory based on quantum error correction. Phys. Rev. Lett. 91(21), 217,901 (2003). doi:10.1103/PhysRevLett.91.217901 CrossRefGoogle Scholar
  22. 22.
    Gisin N., Massar S.: Optimal quantum cloning machines. Phys. Rev. Lett. 79(11), 2153–2156 (1997). doi:10.1103/PhysRevLett.79.2153 ADSCrossRefGoogle Scholar
  23. 23.
    Grassl M., Beth T., Pellizzari T.: Codes for the quantum erasure channel. Phys. Rev. A 56(1), 33–38 (1997). doi:10.1103/PhysRevA.56.33 MathSciNetADSCrossRefGoogle Scholar
  24. 24.
    Holevo A.S.: The capacity of the quantum channel with general signal states. IEEE Trans. Inf. Theory 44, 269–273 (1998)MathSciNetMATHCrossRefGoogle Scholar
  25. 25.
    Horodecki M., Shor P.W., Ruskai M.B.: Entanglement breaking channels. Rev. Math. Phys. 15(6), 629–641 (2003) ArXiv:quant-ph/0302031MathSciNetMATHCrossRefGoogle Scholar
  26. 26.
    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
  27. 27.
    Hsieh M.H., Luo Z., Brun T.: Secret-key-assisted private classical communication capacity over quantum channels. Phys. Rev. A 78(4), 042306 (2008). doi:10.1103/PhysRevA.78.042306 ADSCrossRefGoogle Scholar
  28. 28.
    Hsieh M.H., Wilde M.M.: Public and private communication with a quantum channel and a secret key. Phys. Rev. A 80(2), 022,306 (2009). doi:10.1103/PhysRevA.80.022306 CrossRefGoogle Scholar
  29. 29.
    Hsieh, M.H., Wilde, M.M.: Theory of Quantum Computation, Communication, and Cryptography, Lecture Notes in Computer Science, vol. 5906, Chap. Optimal Trading of Classical Communication, Quantum Communication, and Entanglement, pp. 85–93. Springer (2009)Google 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.
    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
  32. 32.
    King C., Matsumoto K., Nathanson M., Ruskai M.B.: Properties of conjugate channels with applications to additivity and multiplicativity. Markov Process. Relat. Fields 13(2), 391–423 (2007) J. T. Lewis memorial issueMathSciNetMATHGoogle Scholar
  33. 33.
    Korbicz J.K., Almeida M.L., Bae J., Lewenstein M., Acín A. (2008) Structural approximations to positive maps and entanglement-breaking channels. Phys. Rev. A 78(6), 062,105. doi:10.1103/PhysRevA.78.062105
  34. 34.
    Lamas-Linares A., Simon C., Howell J.C., Bouwmeester D.: Experimental quantum cloning of single photons. Science 296, 712–714 (2002)ADSCrossRefGoogle Scholar
  35. 35.
    Lu C.Y., Gao W.B., Zhang J., Zhou X.Q., Yang T., Pan J.W.: Experimental quantum coding against qubit loss error. Proc. Natl. Acad. Sci. USA 105(32), 11,050–11,054 (2008)CrossRefGoogle Scholar
  36. 36.
    Maurer U.: Secret key agreement by public discussion from common information. IEEE Trans. Inf. Theory 39, 733–742 (1993)MATHCrossRefGoogle Scholar
  37. 37.
    Milonni P.W., Hardies M.L.: Photons cannot always be replicated. Phys. Lett. A 92(7), 321–322 (1982)ADSCrossRefGoogle Scholar
  38. 38.
  39. 39.
    Chruściński D., Pytel J., Sarbicki G.: Constructing optimal entanglement witnesses. Phys. Rev. A 80(6), 062,314 (2009). doi:10.1103/PhysRevA.80.062314 Google Scholar
  40. 40.
    Scarani V., Bechmann-Pasquinucci H., Cerf N.J., Dušek M., Lütkenhaus N., Peev M.: The security of practical quantum key distribution. Rev. Modern Phys. 81(3), 1301–1350 (2009). doi:10.1103/RevModPhys.81.1301 ADSCrossRefGoogle Scholar
  41. 41.
    Schumacher B., Westmoreland M.D.: Sending classical information via noisy quantum channels. Phys. Rev. A 56, 131–138 (1997)ADSCrossRefGoogle Scholar
  42. 42.
    Schumacher B., Westmoreland M.D.: Quantum privacy and quantum coherence. Phys. Rev. Lett. 80(25), 5695–5697 (1998). doi:10.1103/PhysRevLett.80.5695 ADSCrossRefGoogle Scholar
  43. 43.
    Shor, P.W.: Additivity of the classical capacity of entanglement-breaking quantum channels. J. Math. Phys. 43(9), 4334–4340 (2002). doi:10.1063/1.1498000. url:http://link.aip.org/link/?JMP/43/4334/1 Google Scholar
  44. 44.
    Simon C., Weihs G., Zeilinger A.: Optimal quantum cloning via stimulated emission. Phys. Rev. Lett. 84(13), 2993–2996 (2000). doi:10.1103/PhysRevLett.84.2993 ADSCrossRefGoogle Scholar
  45. 45.
    Smith G.: Private classical capacity with a symmetric side channel and its application to quantum cryptography. Phys. Rev. A 78(2), 022,306 (2008). doi:10.1103/PhysRevA.78.022306 CrossRefGoogle Scholar
  46. 46.
    Smith G., Renes J.M., Smolin J.A.: Structured codes improve the Bennett-Brassard-84 quantum key rate. Phys. Rev. Lett. 100(17), 170,502 (2008). doi:10.1103/PhysRevLett.100.170502 CrossRefGoogle Scholar
  47. 47.
    Ursin R. et al.: Space-QUEST: experiments with quantum entanglement in space. Europhys. News 40(3), 26–29 (2009) ArXiv:0806.0945ADSCrossRefGoogle Scholar
  48. 48.
    Vernam G.S.: Cipher printing telegraph systems for secret wire and radio telegraphic communications. J. IEEE 55, 109–115 (1926)Google Scholar
  49. 49.
    Wasilewski W., Banaszek K.: Protecting an optical qubit against photon loss. Phys. Rev. A 75(4), 042,316 (2007). doi:10.1103/PhysRevA.75.042316 CrossRefGoogle Scholar
  50. 50.
    Wilde, M.M., Hsieh, M.-H.: The quantum dynamic capacity formula of a quantum channel. Quantum Inf. Process. doi:10.1007/s11128-011-0310-6
  51. 51.
    Wyner A.D.: The wire-tap channel. Bell Syst. Tech. J. 54, 1355–1387 (1975)MathSciNetMATHGoogle Scholar
  52. 52.
    Yard, J.: Simultaneous Classical-Quantum Capacities of Quantum Multiple Access Channels. Ph.D. thesis, Stanford University, Stanford, (2005). Quant-ph/0506050Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.School of Computer ScienceMcGill UniversityMontrealCanada
  2. 2.ERATO-SORST Quantum Computation and Information Project, Japan Science and Technology AgencyTokyoJapan
  3. 3.Statistical LaboratoryUniversity of CambridgeCambridgeUK

Personalised recommendations