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Quantum Information Theory: Results and Open Problems

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Book cover Visions in Mathematics

Part of the book series: Modern Birkhäuser Classics ((MBC))

Abstract

The discipline of information theory was founded by Claude Shannon in a truly remarkable paper [Sh] which laid down the foundations of the subject. We begin with a quote from this paper which is an excellent summary of the main concern of information theory:

The fundamental problem of communication is that of reproducing at one point either exactly or approximately a message selected at another point.

A large part of this paper is included in the paper “Quantum Shannon Theory,” which appeared in the IEEE Information Theory Society Newsletter 50:3 (September 2000), 3–5 and 28–33.

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References

  1. G.G. Amosov, A.S. Holevo, R.F. Werner, On some additivity problems in quantum information theory, LANL e-print math-ph/0003002, available at http://xxx.lanl.gov.

    Google Scholar 

  2. H. Barntjm, J.A. Smolin, B.M. Terhal, Quantum capacity is properly defined without encodings, Phys. Rev. A 58 (1998), 3496–3501.

    Google Scholar 

  3. J.S. Bell, On the Einstein-Podolsky-Rosen paradox, Physics 1 (1964),195–200.

    Google Scholar 

  4. C.H. Bennett, H.J. Bernstein, S. Popescu, B. Schumacher, Concentrating partial entanglement by local operations, Phys. Rev. A 53 (1996), 2046–2052.

    Google Scholar 

  5. C.H. Bennett, G. Brassard, C. Crépeau, R. Jozsa, A. Peres, W.K. Wootters, Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels, Phys. Rev. Lett. 70 (1993), 1895–1899.

    Article  MATH  MathSciNet  Google Scholar 

  6. C.H. Bennett, P.W. Shor, J.A. Smolin, A.V. Thapliyal, Entanglement-assisted classical capacity of noisy quantum channels, Phys. Rev. Lett. 83 (1999), 3081–3084.

    Article  Google Scholar 

  7. C.H. Bennett, P.W. Shor, J.A. Smolin, A.V. Thapliyal, manuscript in preparation.

    Google Scholar 

  8. C.H. Bennett, S.J. Wiesner, Communication via one-and two-particle operators on Einstein-Podolsky-Rosen states, Phys. Rev. Lett. 69 (1992), 2881–2884.

    Article  MATH  MathSciNet  Google Scholar 

  9. E. Biham, M. Boyer, P.O. Boykin, T. Mor, V. Roychowd-hury, A proof of the security of quantum key distribution, in “Proceedings of the Thirty-Second Annual ACM Symposium on Theory of Computing,” ACM Press, New York, (2000), 715–724; longer version LANL e-print quant-ph/9912053, available at http://xxx.lanl.gov.

    Chapter  Google Scholar 

  10. D. DiVincenzo, J.A. Smolin, P.W. Shor, Quantum-channel capacity of very noisy channels, Phys. Rev. A 57 (1998), 830–839.

    Google Scholar 

  11. C.A. Fuchs, A. Peres, personal communication.

    Google Scholar 

  12. J.P. Gordon, Noise at optical frequencies; information theory, in “Quantum Electronics and Coherent Light; Proceedings of the International School of Physics Enrico Fermi, Course XXXI (P.A. Miles, ed.), Academic Press New York (1964), 156–181.

    Google Scholar 

  13. D. Gottesman, An introduction to quantum error correction, LANL e-print quant-ph/0004072, available at http://xxx.lanl.gov.

    Google Scholar 

  14. D.M. Greenburger, M.A. Horne, A. Shimony, A. Zeilinger, Bell’s theorem without inequalities, Am. J. Phys. 58 (1990), 1131–1143.

    Article  Google Scholar 

  15. P. Hausladen, R. Jozsa, B. Schumacher, M. Westmoreland, W.K. Wootters, Classical information capacity of a quantum channel, Phys. Rev. A 54 (1996), 1869–1876.

    MathSciNet  Google Scholar 

  16. K. Hellwig, K. Krauss, Operations and measurements II, Communications in Mathematical Physics 16 (1970), 142–147.

    Article  MATH  MathSciNet  Google Scholar 

  17. A.S. Holevo, Information theoretical aspects of quantum measurements, Probl. Info. Transm. (USSR) 9:2 (1973), 31–42 (in Russian); [translation: A.S. Kholevo, Probl. Info. Transm. 9 (1973), 177–183].

    MATH  MathSciNet  Google Scholar 

  18. A.S. Holevo, The capacity of the quantum channel with general signal states, IEEE Trans. Info. Theory 44 (1998), 269–273.

    Article  MATH  MathSciNet  Google Scholar 

  19. M. Horodecki, P. Horodecki, R. Horodecki, Unified approach to quantum capacities: Towards a quantum noisy coding theorem, LANL e-print quant-ph/0003040, available at http://xxx.lanl.gov.

    Google Scholar 

  20. R. Jozsa, B. Schumacher, A new proof of the quantum noiseless coding theorem, J. Modern Optics 41 (1994), 2343–2349.

    Article  MATH  MathSciNet  Google Scholar 

  21. L.B. Levitin, On the quantum measure of information, in “Proceedings of the Fourth All-Union Conference on Information and Coding Theory, Sec. II”, Tashkent, 1969.

    Google Scholar 

  22. L.B. Levitin, Optimal quantum measurements for pure and mixed states, in “Quantum Communications and Measurement, ” (V.P. Belavkin, O. Hirota, R.L. Hudson, eds.), Plenum Press, New York and London (1995), 439–448.

    Google Scholar 

  23. H.-K. Lo, H.F. Chau, Unconditional security of quantum key distribution over arbitrarily long distances, Science 283 (1999), 2050–2056.

    Article  Google Scholar 

  24. D. Mayers, Unconditional security in quantum cryptography, J. ACM, to appear; also LANL e-print quant-ph/9802025, available at http://xxx.lanl.gov.

    Google Scholar 

  25. J.R. Pierce, The early days of information theory, IEEE Trans. Info. Theory 19 (1973), 3–8.

    Article  MATH  MathSciNet  Google Scholar 

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

    MathSciNet  Google Scholar 

  27. B. Schumacher, M. Westmoreland, Sending classical information via a noisy quantum channel, Phys. Rev. A 56 (1997), 131–138.

    Google Scholar 

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

    MATH  MathSciNet  Google Scholar 

  29. P.W. Shor, J.A. Preskill, Simple proof of security of the BB84 quantum key distribution protocol, Phys. Rev. Lett. 85 (2000), 441–444.

    Article  Google Scholar 

  30. W.K. Wootters, W.H. Zurek, A single quantum cannot be cloned, Nature 299 (1982), 802–803.

    Article  Google Scholar 

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© 2000 Birkhäuser Verlag, Basel

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Shor, P. (2000). Quantum Information Theory: Results and Open Problems. In: Alon, N., Bourgain, J., Connes, A., Gromov, M., Milman, V. (eds) Visions in Mathematics. Modern Birkhäuser Classics. Birkhäuser Basel. https://doi.org/10.1007/978-3-0346-0425-3_9

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