Communications in Mathematical Physics

, Volume 311, Issue 1, pp 97–111 | Cite as

Entanglement can Increase Asymptotic Rates of Zero-Error Classical Communication over Classical Channels

  • Debbie Leung
  • Laura Mancinska
  • William Matthews
  • Maris Ozols
  • Aidan Roy
Article

Abstract

It is known that the number of different classical messages which can be communicated with a single use of a classical channel with zero probability of decoding error can sometimes be increased by using entanglement shared between sender and receiver. It has been an open question to determine whether entanglement can ever increase the zero-error communication rates achievable in the limit of many channel uses. In this paper we show, by explicit examples, that entanglement can indeed increase asymptotic zero-error capacity, even to the extent that it is equal to the normal capacity of the channel.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Bennett C.H., Shor P.W.: Quantum information theory. IEEE Trans. Inf. Th 44, 2724–2742 (1998)MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Horodecki R., Horodecki P., Horodecki M., Horodecki K.: Quantum entanglement. Rev. Mod. Phys. 81(2), 865–942 (2009)MathSciNetADSMATHCrossRefGoogle Scholar
  3. 3.
    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
  4. 4.
    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
  5. 5.
    Bennett C.H., Shor P.W., Smolin J.A., Thapliyal A.V.: Entanglement-assisted capacity of a quantum channel and the reverse Shannon theorem. Information Theory, IEEE Transactions on 48(10), 2637–2655 (2002)MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Polyanskiy Y., Poor H.V., Verdú S.: Channel coding rate in the finite blocklength regime. IEEE Transactions on Information Theory 56(5), 2307–2359 (2010)CrossRefGoogle Scholar
  7. 7.
    Körner J., Orlitsky A.: Zero-error information theory. IEEE Trans. Inf. Theory 44(6), 2207–2229 (1998)MATHCrossRefGoogle Scholar
  8. 8.
    Cubitt, T.S., Leung, D., Matthews, W., Winter, A.: Improving zero-error classical communication with entanglement. Phys. Rev. Lett. 104(23), 230503 (2010)Google Scholar
  9. 9.
    Cubitt, T.S., Leung, D., Matthews, W., Winter, A.: Zero-error channel capacity and simulation assisted by non-local correlations. http://arxiv.org/abs/1003.3195v1 [math. NT], 2010
  10. 10.
    Beigi, S.: Entanglement-assisted zero-error capacity is upper bounded by the Lovász theta function. http://arxiv.org/abs/1002.2488v1 [quant-ph], 2010
  11. 11.
    Duan, R., Severini, S., Winter, A.: Zero-error communication via quantum channels, non-commutative graphs and a quantum Lovász \({\vartheta}\) function. http://arxiv.org/abs/1002.2514v2 [quant-ph], 2010
  12. 12.
    Shannon, C.E.: A mathematical theory of communication. Bell Sys. Tech. J. 27, 379–423, 623–656 (1948)Google Scholar
  13. 13.
    Shannon C.E.: The zero error capacity of a noisy channel. IRE Trans. Inform. Th. 2(3), 8–19 (1956)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Karp, R.: Reducibility among combinatorial problems. In: Miller, R., Thatcher, J. eds. Complexity of Computer Computations. London: Plenum Press, 1972, pp 85–103Google Scholar
  15. 15.
    Lovász L.: On the Shannon capacity of a graph. IEEE Trans. Inf. Theory 25(1), 1–7 (1979)MATHCrossRefGoogle Scholar
  16. 16.
    Haemers W.H.: On some problems of Lovász concerning the Shannon capacity of a graph. IEEE Trans. Inf. Theory 25(2), 231–232 (1979)MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    Haemers, W.H.: An upper bound for the Shannon capacity of a graph. Coll. Math. Soc. János Bolyai 25, 267–272 (1978) available from: http://econpapers.repec.org/RePEc:ner:tilbur:urn:nbn:nl:ui:12-402396, 1978
  18. 18.
    Erdmann K., Wildon M.J.: Introduction to Lie algebras. Springer, Berlin-Heildelberg-Newyork (2006)MATHGoogle Scholar
  19. 19.
    Saniga, M., Planat, M.R.P.: Multiple Qubits as Symplectic Polar Spaces of Order Two. Advanced Studies in Theoretical Physics 1, 1–4 (2007) available from: http://hal.archives-ouvertes.fr/hal-00121565/en/
  20. 20.
    Planat, M., Saniga, M.: On the Pauli graphs of N-qudits. Quantum Information and Computation 8(1-2), 127–146 (2008), available from: http://hal.archives-ouvertes.fr/hal-00127731/en/
  21. 21.
    Havlicek, H., Odehnal, B., Saniga, M.: Factor-Group-Generated Polar Spaces and (Multi-)Qudits. In: Symmetry, Integrability and Geometry: Methods and Applications, 5, 096 (15 pages) (2009), available from: http://hal.archives-ouvertes.fr/hal-00372071/en/, doi:10.3842/SIGMA.2009.096
  22. 22.
    Peeters R.: Orthogonal representations over finite fields and the chromatic number of graphs. Combinatorica 16, 417–431 (1996)MathSciNetMATHCrossRefGoogle Scholar
  23. 23.
    Dye R.H.: Partitions and their stabilizers for line complexes and quadrics. Annali di Matematica Pura ed Applicata 114, 173–194 (1977)MathSciNetMATHCrossRefGoogle Scholar
  24. 24.
    Ball S., Bamberg J., Lavrauw M., Penttila T.: Symplectic spreads. Designs, Codes and Cryptography 32, 9–14 (2004)MathSciNetMATHCrossRefGoogle Scholar
  25. 25.
    Bandyopadhyay S., Boykin O.P., Roychowdhury V., Vatan F.: A new proof for the existence of mutually unbiased bases. Algorithmica 34, 512–528 (2008)MathSciNetGoogle Scholar
  26. 26.
    Lawrence J., Brukner Č., Zeilinger A.: Mutually unbiased binary observable sets on N qubits. Phys. Rev. A 65(3), 032320 (2002)ADSCrossRefGoogle Scholar
  27. 27.
    Cerchiai, B.L., van Geemen, B.: From qubits to E 7. http://arxiv.org/abs/1003.4255v1 [quant-ph], 2010
  28. 28.
    Purbhoo K.: Compression of root systems and the E-sequence. Elec. J. Combinatorics 15(1), R115 (2008)MathSciNetGoogle Scholar
  29. 29.
    Shannon, C.E., Gallager, R.G., Berlekamp, E.R.: Lower bounds to error probability for coding on discrete memoryless channels. I. Information and Control 10(1), 65–103 (1967)Google Scholar
  30. 30.
    Humphreys J.E.: Reflection groups and Coxeter groups. Cambridge University Press, Cambridge (1992)MATHGoogle Scholar
  31. 31.
    Panigrahi P.: The diameters graph of the root system E 8 is uniquely geometrisable. Geometriae Dedicata 78, 121–141 (1999)MathSciNetMATHCrossRefGoogle Scholar
  32. 32.
    Ruuge A.E.: Exceptional and non-crystallographic root systems and the Kochen–Specker theorem. J. Phys. A: Math. Theor. 40(11), 2849–2859 (2007)MathSciNetADSMATHCrossRefGoogle Scholar
  33. 33.
    Prevedel R., Lu Y., Matthews W., Kaltenbaek R., Resch K.J.: Entanglement-enhanced classical communication over a noisy classical channel. Phys. Rev. Lett. 106, 110505 (2011)ADSCrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2012

Authors and Affiliations

  • Debbie Leung
    • 1
  • Laura Mancinska
    • 1
  • William Matthews
    • 1
  • Maris Ozols
    • 1
  • Aidan Roy
    • 1
  1. 1.Institute for Quantum ComputingUniversity of WaterlooWaterlooCanada

Personalised recommendations