Skip to main content
SpringerLink
Log in

Quantumness of quantum ensembles

  • Published:
Theoretical and Mathematical Physics Aims and scope Submit manuscript

Abstract

Quantum ensembles, as generalizations of quantum states, are a universal instrument for describing the physical or informational status in measurement theory and communication theory because of the ubiquitous presence of incomplete information and the necessity of encoding classical messages in quantum states. The interrelations between the constituent states of a quantum ensemble can display more or less quantum characteristics when the involved quantum states do not commute because no single classical basis diagonalizes all these states. This contrasts sharply with the situation of a single quantum state, which is always diagonalizable. To quantify these quantum characteristics and, in particular, to more clearly understand the possibilities of secure data transmission in quantum cryptography, based on certain prototypical quantum ensembles, we introduce some figures of merit quantifying the quantumness of a quantum ensemble, review some existing quantities that are interpretable as measures of quantumness, and investigate their fundamental properties such as subadditivity and concavity. Comparing these measures, we find that different measures can yield different quantumness orderings for quantum ensembles. This reveals the elusive and complex nature of quantum ensembles and shows that no unique measure can describe all the fundamental and subtle properties of quantumness.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. C. A. Fuchs, “Just two nonorthogonal quantum states,” arXiv:quant-ph/9810032v1 (1998).

  2. C. A. Fuchs and M. Sasaki, “The quantumness of a set of quantum states,” arXiv:quant-ph/0302108v1 (2003).

  3. C. A. Fuchs and M. Sasaki, Quantum Inf. Comput., 3, 377–404 (2003).

    MATH  MathSciNet  Google Scholar 

  4. C. A. Fuchs, “On the quantumness of a Hilbert space,” arXiv:quant-ph/0404122v1 (2004).

  5. M. Horodecki, P. Horodecki, R. Horodecki, and M. Piani, Int. J. Quant. Inf., 4, 105–118 (2006).

    Article  MATH  Google Scholar 

  6. J. Eisert and M. B. Plenio, J. Modern Opt., 46, 145–154 (1999); arXiv:quant-ph/9807034v2 (1998).

    ADS  Google Scholar 

  7. S. Virmani and M. B. Plenio, Phys. Lett. A, 268, 31–34 (2000).

    Article  MATH  ADS  MathSciNet  Google Scholar 

  8. C. H. Bennet and G. Brassard, “Quantum cryptography: Public key distribution and coin tossing,” in: Proceedings of IEEE International Conference on Computers, Systems, and Signal Processing (Bangalore, India, December 10–12, 1984), IEEE, New York (1984), p. 175–179.

    Google Scholar 

  9. A. K. Ekert, Phys. Rev. Lett., 67, 661–663 (1991).

    Article  MATH  ADS  MathSciNet  Google Scholar 

  10. C. H. Bennett, Phys. Rev. Lett., 68, 3121–3124 (1992).

    Article  MATH  ADS  MathSciNet  Google Scholar 

  11. S. J. D. Phonex, S. M. Barnett, and A. Chefles, J. Modern Opt., 47, 507–516 (2000).

    ADS  MathSciNet  Google Scholar 

  12. P. W. Shor and J. Preskill, Phys. Rev. Lett., 85, 441–444 (2000); arXiv:quant-ph/0003004v2 (2000).

    Article  ADS  Google Scholar 

  13. N. Gisin, G. Ribordy, W. Tittel, and H. Zbinden, Rev. Modern Phys., 74, 145–195 (2002); arXiv:quant-ph/0101098v2 (2001).

    Article  ADS  Google Scholar 

  14. A. S. Holevo, Problems Inform. Transmission, 9, 110–118 (1973).

    MathSciNet  Google Scholar 

  15. E. Davis, IEEE Trans. Inf. Theory, 24, 596–599 (1978).

    Article  Google Scholar 

  16. A. Peres and W. K. Wootters, Phys. Rev. Lett., 66, 1119–1122 (1991).

    Article  ADS  Google Scholar 

  17. P. Hausladen and W. K. Wootters, J. Modern Opt., 41, 2385–2390 (1994).

    Article  MATH  ADS  MathSciNet  Google Scholar 

  18. J. C. Boileau, K. Tamaki, J. Batuwantudawe, R. Laflamme, and J. M. Renes, Phys. Rev. Lett., 94, 040503 (2005); arXiv:quant-ph/0408085v4 (2004).

    Article  ADS  MathSciNet  Google Scholar 

  19. J. M. Renes, R. Blume-Kohout, A. J. Scott, and C. M. Caves, J. Math. Phys, 45, 2171–2180 (2004); arXiv:quantph/0310075v1 (2003).

    Article  MATH  ADS  MathSciNet  Google Scholar 

  20. A. J. Scott and M. Grassl, J. Math. Phys., 51, 042203 (2010); arXiv:0910.5784v2 [quant-ph] (2009).

    Google Scholar 

  21. D. Bru, Phys. Rev. Lett., 81, 3018–3021 (1998); arXiv:quant-ph/9805019v2 (1998).

    Article  ADS  Google Scholar 

  22. H. Bechmann-Pasquinucci and N. Gisin, Phys. Rev. A, 59, 4238–4248 (1999); arXiv:quant-ph/9807041v2 (1998).

    Article  ADS  MathSciNet  Google Scholar 

  23. Z. Shadman, H. Kampermann, T. Meyer, and D. Bru, Int. J. Quantum Inf., 7, 297–306 (2009).

    Article  MATH  Google Scholar 

  24. W. K. Wootters and B. D. Fields, Ann. Phys., 191, 363–381 (1989).

    Article  ADS  MathSciNet  Google Scholar 

  25. S. Luo, N. Li, and W. Sun, Quant. Inf. Processing, 9, 711–726 (2010).

    Article  MATH  MathSciNet  Google Scholar 

  26. S. Luo, Phys. Rev. A, 77, 022301 (2008).

    Article  ADS  Google Scholar 

  27. H. Ollivier and W. H. Zurek, Phys. Rev. Lett., 88, 017901 (2002).

    Article  ADS  Google Scholar 

  28. L. Henderson and V. Vedral, J. Phys. A, 34, 6899–6905 (2001); arXiv:quant-ph/0105028v1 (2001).

    Article  MATH  ADS  MathSciNet  Google Scholar 

  29. S. Luo, N. Li, and X. Cao, Period. Math. Hungar., 59, 223–237 (2009).

    Article  MATH  MathSciNet  Google Scholar 

  30. R. Jozsa, D. Robb, and W. K. Wootters, Phys. Rev. A, 49, 668–677 (1994).

    Article  ADS  MathSciNet  Google Scholar 

  31. L. B. Levitin, “Optimal quantum measurements for two pure and mixed states,” in: Quantum Communications and Measurement (Univ. of Nottingham, Nottingham, July 10–16, 1994, V. P. Belavkin, O. Hirota, and R. L. Hudson, eds.), Plenum, New York (1995), p. 439–448.

    Google Scholar 

  32. C. A. Fuchs, “Distinguishability and accessible information in quantum theory,” arXiv:quant-ph/9601020v1 (1996).

  33. M. Sasaki, S. M. Barnett, R. Jozsa, M. Osaki, and O. Hirota, Phys. Rev. A, 59, 3325–3335 (1999); arXiv:quantph/9812062v3 (1998).

    Article  ADS  Google Scholar 

  34. J. Mizuno, M. Fujiwara, M. Akiba, T. Kawanishi, S. M. Barnett, and M. Sasaki, Phys. Rev. A, 65, 012315 (2001).

    Google Scholar 

  35. W. K. Wootters and W. H. Zurek, Nature, 299, 802–803 (1982).

    Article  ADS  Google Scholar 

  36. D. Dieks, Phys. Lett. A, 92, 271–272 (1982).

    Article  ADS  Google Scholar 

  37. H. P. Yuen, Phys. Lett. A, 113, 405–407 (1986).

    Article  ADS  MathSciNet  Google Scholar 

  38. H. Barnum, C. M. Caves, C. A. Fuchs, R. Jozsa, and B. Schumacher, Phys. Rev. Lett., 76, 2818–2821 (1996); arXiv:quant-ph/9511010v1 (1995).

    Article  ADS  Google Scholar 

  39. G. Lindblad, Lett. Math. Phys, 47, 189–196 (1999).

    Article  MATH  MathSciNet  Google Scholar 

  40. V. Scarani, S. Iblisdir, N. Gisin, and A. Acín, Rev. Modern Phys, 77, 1225–1256 (2005); arXiv:quant-ph/0511088v1 (2005).

    Article  MATH  ADS  MathSciNet  Google Scholar 

  41. H. Barnum, J. Barrett, M. Leifer, and A. Wilce, Phys. Rev. Lett., 99, 240501 (2007); arXiv:0707.0620v1 [quant-ph] (2007).

    Article  ADS  Google Scholar 

  42. A. Kalev and I. Hen, Phys. Rev. Lett., 100, 210502 (2008); arXiv:0704.1754v3 [quant-ph] (2007).

    Article  ADS  Google Scholar 

  43. D. Bru, Phys. Rev. A, 57, 2368–2378 (1998); arXiv:quant-ph/9705038v3 (1997).

    Article  ADS  Google Scholar 

  44. A. Chefles, Contemp. Phys., 41, 401–424 (2000); arXiv:quant-ph/0010114v1 (2000).

    Article  ADS  Google Scholar 

  45. S. M. Barnett and S. Croke, “Quantum state discrimination,” arXiv:0810.1970v1 [quant-ph] (2008).

  46. K. M. R. Audenaert, C. A. Fuchs, C. King, and A. Winter, Quantum Inf. Comput., 4, 1–11 (2004).

    MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, People’s Republic of China

    Shunlong Luo, Nan Li & Shuangshuang Fu

Authors
  1. Shunlong Luo
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Nan Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Shuangshuang Fu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Shunlong Luo.

Additional information

Prepared from an English manuscript submitted by the authors; for the Russian version, see Teoreticheskaya i Matematicheskaya Fizika, Vol. 169, No. 3, pp. 413–430, December, 2011.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Luo, S., Li, N. & Fu, S. Quantumness of quantum ensembles. Theor Math Phys 169, 1724–1739 (2011). https://doi.org/10.1007/s11232-011-0147-2

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11232-011-0147-2

Keywords

Search

Navigation