Partial quantum information

Abstract

Information—be it classical1 or quantum2—is measured by the amount of communication needed to convey it. In the classical case, if the receiver has some prior information about the messages being conveyed, less communication is needed3. Here we explore the concept of prior quantum information: given an unknown quantum state distributed over two systems, we determine how much quantum communication is needed to transfer the full state to one system. This communication measures the partial information one system needs, conditioned on its prior information. We find that it is given by the conditional entropy—a quantity that was known previously, but lacked an operational meaning. In the classical case, partial information must always be positive, but we find that in the quantum world this physical quantity can be negative. If the partial information is positive, its sender needs to communicate this number of quantum bits to the receiver; if it is negative, then sender and receiver instead gain the corresponding potential for future quantum communication. We introduce a protocol that we term ‘quantum state merging’ which optimally transfers partial information. We show how it enables a systematic understanding of quantum network theory, and discuss several important applications including distributed compression, noiseless coding with side information, multiple access channels and assisted entanglement distillation.

Access options

Rent or Buy article

Get time limited or full article access on ReadCube.

from$8.99

All prices are NET prices.

Figure 1: Diagrammatic representation of the process of state merging.
Figure 2: The rate region for distributed compression by two parties with individual rates RA and RB.

References

  1. 1

    Shannon, C. E. A mathematical theory of communication. Bell Syst. Tech. J. 27, 379–423, 623–656 (1948)

    MathSciNet  Article  Google Scholar 

  2. 2

    Schumacher, B. Quantum coding. Phys. Rev. A 51, 2738–2747 (1995)

    ADS  MathSciNet  CAS  Article  Google Scholar 

  3. 3

    Slepian, D. & Wolf, J. K. Noiseless coding of correlated information sources. IEEE Trans. Inf. Theory 19, 461–480 (1971)

    MathSciNet  MATH  Google Scholar 

  4. 4

    Wehrl, A. General properties of entropy. Rev. Mod. Phys. 50, 221–260 (1978)

    ADS  MathSciNet  Article  Google Scholar 

  5. 5

    Horodecki, R. & Horodecki, P. Quantum redundancies and local realism. Phys. Lett. A 194, 147–152 (1994)

    ADS  MathSciNet  CAS  Article  Google Scholar 

  6. 6

    Cerf, N. J. & Adami, C. Negative entropy and information in quantum mechanics. Phys. Rev. Lett. 79, 5194–5197 (1997)

    ADS  MathSciNet  CAS  Article  Google Scholar 

  7. 7

    Einstein, A., Podolsky, B. & Rosen, N. Can quantum-mechanical description of physical reality be considered complete? Phys. Rev. 47, 777–780 (1935)

    ADS  CAS  Article  Google Scholar 

  8. 8

    Bennett, C. H. et al. Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels. Phys. Rev. Lett. 70, 1895–1899 (1993)

    ADS  MathSciNet  CAS  Article  Google Scholar 

  9. 9

    Schrödinger, E. Die gegenwärtige Situation der Quantenmechanik. Naturwissenschaften 23, 807–821, 823–828, 844–849 (1935)

    ADS  Article  Google Scholar 

  10. 10

    Groisman, B., Popescu, S. & Winter, A. On the quantum, classical and total amount of correlations in a quantum state. Preprint at http://arXiv.org/quant-ph/0410091 (2004).

  11. 11

    Horodecki, M. et al. Local versus non-local information in quantum information theory: formalism and phenomena. Phys. Rev. A 71, 062307 (2005)

    ADS  Article  Google Scholar 

  12. 12

    Ahn, C., Doherty, A., Hayden, P. & Winter, A. On the distributed compression of quantum information. Preprint at http://arXiv.org/quant-ph/0403042 (2004).

  13. 13

    Wyner, A. On source coding with the side information at the decoder. IEEE Trans. Inf. Theory 21, 294–300 (1975)

    MathSciNet  Article  Google Scholar 

  14. 14

    Terhal, B. M., Horodecki, M., DiVincenzo, D. P. & Leung, D. The entanglement of purification. J. Math. Phys. 43, 4286–4298 (2002)

    ADS  MathSciNet  CAS  Article  Google Scholar 

  15. 15

    Schumacher, B. & Nielsen, M. A. Quantum data processing and error correction. Phys. Rev. A 54, 2629–2635 (1996)

    ADS  MathSciNet  CAS  Article  Google Scholar 

  16. 16

    Shor, P. The quantum channel capacity and coherent information. MSRI Workshop on Quantum Computation (Berkeley, 2002); lecture notes available at http://www.msri.org/publications/ln/msri/2002/quantumcrypto/shor/1.

  17. 17

    Lloyd, S. The capacity of the noisy quantum channel. Phys. Rev. A 55, 1613–1622 (1997)

    ADS  MathSciNet  CAS  Article  Google Scholar 

  18. 18

    Devetak, I . The private classical capacity and quantum capacity of a quantum channel. IEEE Trans. Inf. Theory 51(1), 44–55 (2005)

    MathSciNet  Article  Google Scholar 

  19. 19

    Yard, J., Devetak, I. & Hayden, P. Capacity theorems for quantum multiple access channels—Part I: classical-quantum and quantum-quantum capacity regions (2005). Preprint at http://arXiv.org/quant-ph/0501045 (2005).

  20. 20

    DiVincenzo, D. P., et al. in Proc. 1st NASA Int. Conf. on Quantum Computing and Quantum Communication (ed. Williams, C. P.) LNCS 1509 247–257 (Springer, London, 1999)

    Google Scholar 

  21. 21

    Smolin, J. A., Verstraete, F. & Winter, A. Entanglement of assistance and multi-partite state distillation. Preprint at http://arXiv.org/quant-ph/0505038 (2005).

  22. 22

    Lieb, E. H. & Ruskai, M. B. Proof of the strong subadditivity of quantum-mechanical entropy with an appendix by B. Simon. J. Math. Phys. 14, 1938–1941 (1973)

    ADS  Article  Google Scholar 

  23. 23

    Devetak, I. & Winter, A. Relating quantum privacy and quantum coherence: an operational approach. Phys. Rev. Lett. 93, 080501 (2004)

    ADS  CAS  Article  Google Scholar 

Download references

Acknowledgements

This work was done at the Isaac Newton Institute (Cambridge) in August–December 2004 and we are grateful for the institute's hospitality. We thank W. Unruh for comments on an earlier draft of this paper. We acknowledge the support of EC grants RESQ, QUPRODIS and PROSECCO. M.H. was additionally supported by the Polish Ministry of Scientific Research and Information Technology, J.O. by the Cambridge-MIT Institute and Newton Trust, and A.W. by EPSRC's “IRC QIP”.

Author information

Affiliations

Authors

Corresponding author

Correspondence to Jonathan Oppenheim.

Ethics declarations

Competing interests

Reprints and permissions information is available at npg.nature.com/reprintsandpermissions. The authors declare no competing financial interests.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Horodecki, M., Oppenheim, J. & Winter, A. Partial quantum information. Nature 436, 673–676 (2005). https://doi.org/10.1038/nature03909

Download citation

Further reading

Comments

By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate.

Search

Nature Briefing

Sign up for the Nature Briefing newsletter — what matters in science, free to your inbox daily.

Get the most important science stories of the day, free in your inbox. Sign up for Nature Briefing