Quantum Information Processing

, Volume 9, Issue 5, pp 509–540 | Cite as

Quantum convolutional coding with shared entanglement: general structure

  • Mark M. WildeEmail author
  • Todd A. Brun


We present a general theory of entanglement-assisted quantum convolutional coding. The codes have a convolutional or memory structure, they assume that the sender and receiver share noiseless entanglement prior to quantum communication, and they are not restricted to possess the Calderbank–Shor–Steane structure as in previous work. We provide two significant advances for quantum convolutional coding theory. We first show how to “expand” a given set of quantum convolutional generators. This expansion step acts as a preprocessor for a polynomial symplectic Gram–Schmidt orthogonalization procedure that simplifies the commutation relations of the expanded generators to be the same as those of entangled Bell states (ebits) and ancilla qubits. The above two steps produce a set of generators with equivalent error-correcting properties to those of the original generators. We then demonstrate how to perform online encoding and decoding for a stream of information qubits, halves of ebits, and ancilla qubits. The upshot of our theory is that the quantum code designer can engineer quantum convolutional codes with desirable error-correcting properties without having to worry about the commutation relations of these generators.


Quantum convolutional codes Entanglement-assisted quantum convolutional codes Quantum information theory Entanglement-assisted quantum codes 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Ollivier H., Tillich J.-P.: Description of a quantum convolutional code. Phys. Rev. Lett. 91(17), 177902 (2003)CrossRefADSPubMedGoogle Scholar
  2. 2.
    Ollivier, H., Tillich, J.-P.: Quantum convolutional codes: fundamentals. arXiv:quant-ph/0401134 (2004)Google Scholar
  3. 3.
    Grassl, M., Rötteler, M.: Noncatastrophic encoders and encoder inverses for quantum convolutional codes. In: IEEE International Symposium on Information Theory (quant-ph/0602129) (2006)Google Scholar
  4. 4.
    Grassl, M., Rötteler, M.: Quantum convolutional codes: encoders and structural properties. In: Proceedings of the Forty-Fourth Annual Allerton Conference, pp. 510–519 (2006)Google Scholar
  5. 5.
    Grassl, M., Rötteler, M.: Constructions of quantum convolutional codes. In: Proceedings of the IEEE International Symposium on Information Theory, pp. 816–820 (2007)Google Scholar
  6. 6.
    Forney, G.D., Guha, S.: Simple rate-1/3 convolutional and tail-biting quantum error-correcting codes. In: IEEE International Symposium on Information Theory (arXiv:quant-ph/0501099) (2005)Google Scholar
  7. 7.
    Forney G.D., Grassl M., Guha S.: Convolutional and tail-biting quantum error-correcting codes. IEEE Trans. Inf. Theory 53, 865–880 (2007)CrossRefMathSciNetGoogle Scholar
  8. 8.
    Aly, S.A., Grassl, M., Klappenecker, A., Roetteler, M., Sarvepalli, P.K.: Quantum convolutional BCH codes. In: 10th Canadian Workshop on Information Theory (arXiv:quant-ph/0703113), pp. 180–183 (2007)Google Scholar
  9. 9.
    Aly, S.A., Klappenecker, A., Sarvepalli, P.K.: Quantum convolutional codes derived from Reed-Solomon and Reed-Muller codes. arXiv:quant-ph/0701037 (2007)Google Scholar
  10. 10.
    Wilde, M.M., Krovi, H., Brun, T.A.: Convolutional entanglement distillation. To appear in the International Symposium on Information Theory, arXiv:0708.3699, Austin, Texas, USA June (2010)Google Scholar
  11. 11.
    Wilde M.M., Brun T.A.: Entanglement-assisted quantum convolutional coding. Phys. Rev. A 81, 042333 (2010)CrossRefADSGoogle Scholar
  12. 12.
    Wilde, M.M., Brun, T.A.: Unified quantum convolutional coding. In: Proceedings of the IEEE International Symposium on Information Theory (arXiv:0801.0821), pp. 359–363. Toronto, ON, Canada, July 2008Google Scholar
  13. 13.
    Lloyd S.: Capacity of the noisy quantum channel. Phys. Rev. A 55(3), 1613–1622 (1997)CrossRefMathSciNetADSGoogle Scholar
  14. 14.
    Shor, P.W.: The quantum channel capacity and coherent information. In: Lecture Notes, MSRI Workshop on Quantum Computation (2002)Google Scholar
  15. 15.
    Devetak I.: The private classical capacity and quantum capacity of a quantum channel. IEEE Trans. Inf. Theory 51, 44–55 (2005)CrossRefMathSciNetGoogle Scholar
  16. 16.
    Hayden P., Horodecki M., Winter A., Yard J.: A decoupling approach to the quantum capacity. Open Syst. Inf. Dyn. 15, 7–19 (2008)CrossRefMathSciNetzbMATHGoogle Scholar
  17. 17.
    Klesse R.: A random coding based proof for the quantum coding theorem. Open Syst. Inf. Dyn. 15, 21–45 (2008)CrossRefMathSciNetzbMATHGoogle Scholar
  18. 18.
    Horodecki M., Lloyd S., Winter A.: Quantum coding theorem from privacy and distinguishability. Open Syst. Inf. Dyn. 15, 47–69 (2008)CrossRefMathSciNetzbMATHGoogle Scholar
  19. 19.
    Hayden P., Shor P.W., Winter A.: Random quantum codes from Gaussian ensembles and an uncertainty relation. Open Syst. Inf. Dyn. 15, 71–89 (2008)CrossRefMathSciNetzbMATHGoogle Scholar
  20. 20.
    Poulin D., Tillich J.-P., Ollivier H.: Quantum serial turbo-codes. IEEE Trans. Inf. Theory 55(6), 2776–2798 (2009)CrossRefMathSciNetGoogle Scholar
  21. 21.
    Calderbank A.R., Rains E.M., Shor P.W., Sloane N.J.A.: Quantum error correction via codes over GF(4). IEEE Trans. Inf. Theory 44, 1369–1387 (1998)CrossRefMathSciNetzbMATHGoogle Scholar
  22. 22.
    Nielsen M.A., Chuang I.L.: Quantum Computation and Quantum Information. Cambridge University Press, Cambridge (2000)zbMATHGoogle Scholar
  23. 23.
    Brun T.A., Devetak I., Hsieh M.-H.: Correcting quantum errors with entanglement. Science 314(5798), 436–439 (2006)CrossRefMathSciNetADSPubMedGoogle Scholar
  24. 24.
    Brun, T.A., Devetak, I., Hsieh, M.-H.: Catalytic quantum error correction. arXiv:quant-ph/0608027, August (2006)Google Scholar
  25. 25.
    Grassl, M.: Convolutional and block quantum error-correcting codes. In: IEEE Information Theory Workshop, pp. 144–148, Chengdu, October (2006)Google Scholar
  26. 26.
    Johannesson R., Zigangirov K.Sh.: Fundamentals of Convolutional Coding. Wiley-IEEE Press, (1999)Google Scholar
  27. 27.
    Gottesman, D.: Stabilizer Codes and Quantum Error Correction. PhD thesis, California Institute of Technology (1997)Google Scholar
  28. 28.
    Shaw B., Wilde M.M., Oreshkov O., Kremsky I., Lidar D.: Encoding one logical qubit into six physical qubits. Phys. Rev. A 78, 012337 (2008)CrossRefADSGoogle Scholar
  29. 29.
    Wilde M.M., Brun T.A.: Optimal entanglement formulas for entanglement-assisted quantum coding. Phys. Rev. A 77, 064302 (2008)CrossRefADSGoogle Scholar
  30. 30.
    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)CrossRefMathSciNetADSPubMedzbMATHGoogle Scholar
  31. 31.
    Wilde M.M., Brun T.A.: Extra shared entanglement reduces memory demand in quantum convolutional coding. Phys. Rev. A 79(3), 032313 (2009)CrossRefMathSciNetADSGoogle Scholar
  32. 32.
    Kremsky I., Hsieh M.-H., Brun T.A.: Classical enhancement of quantum-error-correcting codes. Phys. Rev. A 78(1), 012341 (2008)CrossRefADSGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.School of Computer ScienceMcGill UniversityMontrealCanada
  2. 2.Center for Quantum Information Science and Technology and the Communication Sciences Institute of the Ming Hsieh Department of Electrical EngineeringUniversity of Southern CaliforniaLos AngelesUSA

Personalised recommendations