Source Coding in Quantum Systems

  • Masahito HayashiEmail author
Part of the Graduate Texts in Physics book series (GTP)


Nowadays, data compression software has become an indispensable tool for current network system. Why is such a compression possible? Information commonly possesses redundancies. In other words, information possesses some regularity. If one randomly types letters of the alphabet, it is highly unlikely letters that form a meaningful sentence or program. Imagine that we are assigned a task of communicating a sequence of 1000 binary digits via telephone. Assume that the binary digits satisfies the following rule: The 2nth and \((2n+1)\)th digits of this sequence are the same. Naturally, we would not read out all 1000 digits of the sequence; we would first explain that the 2nth and \((2n+1)\)th digits are the same, and then read out the even-numbered (or odd-numbered) digits. We may even check whether there is any further structure in the sequence. In this way, compression software works by changing the input sequence of letters (or numbers) into another sequence of letters that can reproduce the original sequence, thereby reducing the necessary storage. The compression process may therefore be regarded as an encoding. This procedure is called source coding in order to distinguish it from the channel coding examined in Chap.  4. Applying this idea to the quantum scenario, the presence of any redundant information in a quantum system may be similarly compressed to a smaller quantum memory for storage or communication. However, in contrast to the classical case, we have at least two distinct scenarios. The task of the first scenario is saving memory in a quantum computer. This will be relevant when quantum computers are used in practice. In this case, a given quantum state is converted into a state on a system of lower size (dimension). The original state must then be recoverable from the compressed state. Note that the encoder does not know what state is to be compressed. The task in the second scenario is to save the quantum system to be sent for quantum cryptography. In this case, the sender knows what state to be sent. This provides the encoder with more options for compression. In the decompression stage, there is no difference between the first and second scenarios, since their tasks are conversions from one quantum system to another. In this chapter, the two scenarios of compression outlined above are discussed in detail.


Compression Rate Quantum Memory Classical Channel Direct Part Entropy Rate 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. 1.
    C.E. Shannon, A mathematical theory of communication. Bell Syst. Tech. J. 27, 623–656 (1948)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    T.S. Han, K. Kobayashi, Mathematics of Information and Encoding (American Mathematical Society, 2002) (originally appeared in Japanese in 1999)Google Scholar
  3. 3.
    B. Schumacher, Quantum coding. Phys. Rev. A 51, 2738–2747 (1995)ADSMathSciNetCrossRefGoogle Scholar
  4. 4.
    R. Jozsa, B. Schumacher, A new proof of the quantum noiseless coding theorem. J. Mod. Opt. 41(12), 2343–2349 (1994)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Y. Mitsumori, J.A. Vaccaro, S.M. Barnett, E. Andersson, A. Hasegawa, M. Takeoka, M. Sasaki, Experimental demonstration of quantum source coding. Phys. Rev. Lett. 91, 217902 (2003)ADSCrossRefzbMATHGoogle Scholar
  6. 6.
    M. Koashi, N. Imoto, Quantum information is incompressible without errors. Phys. Rev. Lett. 89, 097904 (2002)ADSCrossRefGoogle Scholar
  7. 7.
    L.D. Davisson, Comments on sequence time coding for data compression. Proc. IEEE 54, 2010 (1966)CrossRefGoogle Scholar
  8. 8.
    T.J. Lynch, Sequence time coding for data compression. Proc. IEEE 54, 1490–1491 (1966)CrossRefGoogle Scholar
  9. 9.
    M. Hayashi, K. Matsumoto, Quantum universal variable-length source coding. Phys. Rev. A 66, 022311 (2002)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    M. Hayashi, K. Matsumoto, Simple construction of quantum universal variable-length source coding. Quant. Inf. Comput. 2, Special Issue, 519–529 (2002)Google Scholar
  11. 11.
    I. Devetak, A. Winter, Classical data compression with quantum side information. Phys. Rev. A 68, 042301 (2003)ADSCrossRefGoogle Scholar
  12. 12.
    P. Hayden, R. Jozsa, A. Winter, Trading quantum for classical resources in quantum data compression. J. Math. Phys. 43, 4404–4444 (2002)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    M. Hayashi, Exponents of quantum fixed-length pure state source coding. Phys. Rev. A 66, 032321 (2002)ADSCrossRefGoogle Scholar
  14. 14.
    T.S. Han, Folklore in source coding: information-spectrum approach. IEEE Trans. Inf. Theory 51(2), 747–753 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    M. Hayashi, Second-order asymptotics in fixed-length source coding and intrinsic randomness. IEEE Trans. Inf. Theory 54, 4619–4637 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    I. Csiszár, J. Körner, Information Theory: Coding Theorems for Discrete Memoryless Systems (Academic, 1981)Google Scholar
  17. 17.
    R. Jozsa, M. Horodecki, P. Horodecki, R. Horodecki, Universal quantum information compression. Phys. Rev. Lett. 81, 1714 (1998)ADSCrossRefzbMATHGoogle Scholar
  18. 18.
    M. Koashi, N. Imoto, Compressibility of mixed-state signals. Phys. Rev. Lett. 87, 017902 (2001)ADSCrossRefGoogle Scholar
  19. 19.
    M. Horodecki, Limits for compression of quantum information carried by ensembles of mixed states. Phys. Rev. A 57, 3364–3369 (1998)ADSMathSciNetCrossRefGoogle Scholar
  20. 20.
    M. Horodecki, Optimal compression for mixed signal states. Phys. Rev. A 61, 052309 (2000)ADSMathSciNetCrossRefGoogle Scholar
  21. 21.
    H.-K. Lo, S. Popescu, Concentrating entanglement by local actions: beyond mean values. Phys. Rev. A 63, 022301 (2001)ADSCrossRefGoogle Scholar
  22. 22.
    C.H. Bennett, P.W. Shor, J.A. Smolin, A.V. Thapliyal, Entanglement-assisted classical capacity of noisy quantum channels. Phys. Rev. Lett. 83, 3081 (1999)ADSCrossRefGoogle Scholar
  23. 23.
    W. Dür, G. Vidal, J.I. Cirac, Visible compression of commuting mixed state. Phys. Rev. A 64, 022308 (2001)ADSCrossRefGoogle Scholar
  24. 24.
    C.H. Bennett, P.W. Shor, J.A. Smolin, A.V. Thapliyal, Entanglement-assisted capacity of a quantum channel and the reverse Shannon theorem. IEEE Trans. Inf. Theory 48(10), 2637–2655 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    A. Winter, "Extrinsic" and "intrinsic" data in quantum measurements: asymptotic convex decomposition of positive operator valued measures. Commun. Math. Phys. 244(1), 157–185 (2004)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    A. Winter, S. Massar, Compression of quantum measurement operations. Phys. Rev. A 64, 012311 (2001)ADSCrossRefGoogle Scholar
  27. 27.
    H. Barnum, C.A. Fuchs, R. Jozsa, B. Schumacher, A general fidelity limit for quantum channels. Phys. Rev. A 54, 4707–4711 (1996)ADSCrossRefGoogle Scholar
  28. 28.
    H. Barnum, C.M. Caves, C.A. Fuchs, R. Jozsa, B. Schumacher, On quantum coding for ensembles of mixed states. J. Phys. A Math. Gen. 34, 6767–6785 (2001)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    A. Winter, Schumacher’s quantum coding revisited. Preprint 99–034, Sonder forschungsbereich 343. Diskrete Strukturen in der Mathematik Universität Bielefeld (1999)Google Scholar
  30. 30.
    R. Jozsa, S. Presnell, Universal quantum information compression and degrees of prior knowledge. Proc. R. Soc. Lond. A 459, 3061–3077 (2003)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    C.H. Bennett, A.W. Harrow, S. Lloyd, Universal quantum data compression via nondestructive tomography. Phys. Rev. A 73, 032336 (2006)ADSCrossRefGoogle Scholar
  32. 32.
    M. Hayashi, Universal approximation of multi-copy states and universal quantum lossless data compression. Commun. Math. Phys. 293(1), 171–183 (2010)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    D. Petz, M. Mosonyi, Stationary quantum source coding. J. Math. Phys. 42, 48574864 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    I. Bjelaković, A. Szkoła, The data compression theorem for ergodic quantum information sources. Quant. Inf. Process. 4, 49–63 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    N. Datta, Y. Suhov, Data compression limit for an information source of interacting qubits. Quant. Inf. Process. 1(4), 257–281 (2002)MathSciNetCrossRefGoogle Scholar
  36. 36.
    I. Bjelaković, T. Kruger, R. Siegmund-Schultze, A. Szkoła, The Shannon-McMillan theorem for ergodic quantum lattice systems. Invent. Math. 155, 203–222 (2004)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    H. Nagaoka, M. Hayashi, An information-spectrum approach to classical and quantum hypothesis testing. IEEE Trans. Inf. Theory 53, 534–549 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    A. Kaltchenko, E.-H. Yang, Universal compression of ergodic quantum sources. Quant. Inf. Comput. 3, 359–375 (2003)MathSciNetzbMATHGoogle Scholar
  39. 39.
    R. Jozsa, Quantum noiseless coding of mixed states, in Talk given at 3rd Santa Fe Workshop on Complexity, Entropy, and the Physics of Information, May 1994Google Scholar
  40. 40.
    C.H. Bennett, A. Winter, Private CommunicationGoogle Scholar
  41. 41.
    A.D. Wyner, The common information of two dependent random variables. IEEE Trans. Inf. Theory 21, 163–179 (1975)MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    D. Slepian, J.K. Wolf, Noiseless coding of correlated information sources. IEEE Trans. Inf. Theory 19, 471 (1973)MathSciNetCrossRefzbMATHGoogle Scholar
  43. 43.
    C. Ahn, A. Doherty, P. Hayden, A. Winter, On the distributed compression of quantum information. IEEE Trans. Inform. Theory 52, 4349–4357 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  44. 44.
    A. Abeyesinghe, I. Devetak, P. Hayden, A. Winter, The mother of all protocols: Restructuring quantum information’s family tree. Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, rspa20090202 (2009)Google Scholar
  45. 45.
    M. Hayashi, Optimal Visible Compression Rate For Mixed States Is Determined By Entanglement Purification. Phy. Rev. A Rapid Commun. 73, 060301(R) (2006)CrossRefGoogle Scholar
  46. 46.
    M. Tomamochel, M. Hayashi, A Hierarchy of Information Quantities for Finite Block Length Analysis of Quantum Tasks. IEEE Trans. Inf. Theory 59(11), 7693–7710 (2013)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  1. 1.Graduate School of MathematicsNagoya UniversityNagoyaJapan

Personalised recommendations