Rényi entropy uncertainty relation for successive projective measurements | Quantum Information Processing Skip to main content

Advertisement

Springer Nature Link
Log in

Rényi entropy uncertainty relation for successive projective measurements

  • Published:
Quantum Information Processing Aims and scope Submit manuscript

Abstract

We investigate the uncertainty principle for two successive projective measurements in terms of Rényi entropy based on a single quantum system. Our results cover a large family of the entropy (including the Shannon entropy) uncertainty relations with a lower optimal bound. We compare our relation with other formulations of the uncertainty principle in two-spin observables measured on a pure quantum state of qubit. It is shown that the low bound of our uncertainty relation has better tightness.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
¥17,985 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price includes VAT (Japan)

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4

Similar content being viewed by others

Explore related subjects

Discover the latest articles, news and stories from top researchers in related subjects.

References

  1. Heisenberg, W.J.Z.: Über den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik. Z. Phys. 43, 172 (1927)

    Article  ADS  MATH  Google Scholar 

  2. Robertson, H.P.: The uncertainty principle. Phys. Rev. 34, 163 (1929)

    Article  ADS  Google Scholar 

  3. Busch, P., Heinonen, T., Lahti, P.: Heisenbergs uncertainty principle. Phys. Rep. 452, 155 (2007)

    Article  ADS  Google Scholar 

  4. Berta, M., Christandl, M., Colbeck, R., Renes, J.M., Renner, R.: The uncertainty principle in the presence of quantum memory. Nat. Phys. 6, 659 (2010)

    Article  Google Scholar 

  5. Oppenheim, J., Wehner, S.: The uncertainty principle determines the nonlocality of quantum mechanics. Science 330, 1072 (2010)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  6. Busch, P., Lahti, P., Werner, R.F.: Heisenberg uncertainty for qubit measurements. Phys. Rev. A 89, 012129 (2014)

    Article  ADS  Google Scholar 

  7. Ozawa, M.: Universally valid reformulation of the Heisenberg uncertainty principle on noise and disturbance in measurement. Phys. Rev. A 67, 042105 (2003)

    Article  ADS  Google Scholar 

  8. Erhart, J., Spona, S., Sulyok, G., Badurek, G., Ozawa, M., Yuji, H.: Experimental demonstration of a universally valid error-disturbance uncertainty relation in spin measurements. Nat. Phys. 8, 185 (2012)

    Article  Google Scholar 

  9. Kaneda, F., Baek, S.-Y., Ozawa, M., Edamatsu, K.: Experimental test of error-disturbance uncertainty relations by weak measurement. Phy. Rev. Lett. 112, 020402 (2014)

    Article  ADS  Google Scholar 

  10. Rozema, L.A., Darabi, A., Mahler, D.H., Hayat, A., Soudagar, Y., Steinberg, A.M.: Violation of Heisenberg’s measurement-disturbance relationship by weak measurements. Phys. Rev. Lett. 109, 100404 (2012)

    Article  ADS  Google Scholar 

  11. Baek, S.Y., Kaneda, F., Ozawa, M., Edamatsu, K.: Experimental violation and reformulation of the Heisenberg’s error-disturbance uncertainty relation. Sci. Rep. 3, 2221 (2013)

    ADS  Google Scholar 

  12. Sulyok, G., Sponar, S., Erhart, J., Badurek, G., Ozawa, M., Hasegawa, Y.: Violation of Heisenberg’s error-disturbance uncertainty relation in neutron-spin measurements. Phys. Rev. A 88, 022110 (2013)

    Article  ADS  Google Scholar 

  13. Ringbauer, M., Biggerstaff, D.N., Broome, M.A., Fedrizzi, A., Branciard, C., White, A.G.: Experimental joint quantum measurements with minimum uncertainty. Phys. Rev. Lett. 112, 020401 (2014)

    Article  ADS  Google Scholar 

  14. Deutsch, D.: Uncertainty in quantum measurements. Phys. Rev. Lett. 50, 631 (1983)

    Article  ADS  MathSciNet  Google Scholar 

  15. Bialynicki-Birula, I., Mycielski, J.: Uncertainty relations for information entropy in wave mechanics. Commun. Math. Phys. 44, 129 (1975)

    Article  ADS  MathSciNet  Google Scholar 

  16. Kraus, K.: Complementary observables and uncertainty relation. Phys. Rev. D 35, 3070 (1987)

    Article  ADS  MathSciNet  Google Scholar 

  17. Maassen, H., Uffink, J.B.M.: Generalized entropic uncertainty relations. Phys. Rev. Lett. 60, 1103 (1988)

    Article  ADS  MathSciNet  Google Scholar 

  18. Coles, P.J., Piani, M.: Improved entropic uncertainty relations and information exclusion relations. Phys. Rev. A 89, 022112 (2014)

    Article  ADS  Google Scholar 

  19. Rudnicki, Ł., Puchała, Z., Źyczkowski, K.: Strong majorization entropic uncertainty relations. Phys. Rev. A 89, 052115 (2014)

    Article  ADS  Google Scholar 

  20. Tomamichel, M., Renner, R.: Uncertainty relation for smooth entropies. Phys. Rev. Lett. 106, 110506 (2011)

    Article  ADS  Google Scholar 

  21. Coles, P.J., Colbeck, R., Yu, L., Zwolak, M.: Uncertainty relations from simple entropic properties. Phys. Rev. Lett. 108, 210405 (2012)

    Article  ADS  Google Scholar 

  22. Bialynicki-Birula, I.: Formulation of the uncertainty relations in terms of the Rényi entropies. Phys. Rev. A. 74, 052101 (2006)

    Article  ADS  MathSciNet  Google Scholar 

  23. Zozor, S., Vignat, C.: On classes of non-Gaussian asymptotic minimizers in entropic uncertainty principles. Phys. A 375, 499 (2007)

    Article  Google Scholar 

  24. Zozor, S., Portesi, M., Vignat, C.: Some extensions of the uncertainty principle. Phys. A 387, 4800 (2008)

    Article  MathSciNet  Google Scholar 

  25. Rastegin, A.E.: Rényi formulation of the entropic uncertainty principle for POVMs. J. Phys. A 43, 155302 (2010)

    Article  ADS  MathSciNet  Google Scholar 

  26. Luis, A.: Effect of fluctuation measures on the uncertainty relations between two observables: different measures lead to opposite conclusions. Phys. Rev. A 84, 034101 (2011)

    Article  ADS  Google Scholar 

  27. Rastegin, A.E.: Uncertainty and certainty relations for Pauli observables in terms of Rényi entropies of order \(\alpha \in (0; 1]\). Commun. Theor. Phys. 61, 293 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  28. Bosyk, G.M., Portesi, M., Plastino, A.: Collision entropy and optimal uncertainty. Phys. Rev. A 85, 012108 (2012)

    Article  ADS  Google Scholar 

  29. Ghirardi, G.C., Marinatto, L., Romano, R.: An optimal entropic uncertainty relation in a two-dimensional Hilbert space. Phys. Lett. A 317, 32 (2003)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  30. Wilk, G., Włodarczyk, Z.: Uncertainty relations in terms of the Tsallis entropy. Phys. Rev. A 79, 062108 (2009)

    Article  ADS  MathSciNet  Google Scholar 

  31. Bialynicki-Birula, I., Rudnicki, Ł.: Comment on “Uncertainty relations in terms of the Tsallis entropy”. Phys. Rev. A 81, 026101 (2010)

    Article  ADS  Google Scholar 

  32. Rastegin, A.E.: Uncertainty and certainty relations for complementary qubit observables in terms of Tsallis entropies. Quantum Inf. Process. 12, 2947 (2013)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  33. Rastegin, A.E.: Notes on entropic uncertainty relations beyond the scope of Rieszs theorem. Int. J. Theor. Phys. 51, 1300 (2011)

    Article  MathSciNet  Google Scholar 

  34. Rastegina, A.E.: Entropic formulation of the uncertainty principle for the number and annihilation operators. Phys. Scr. 84, 057001 (2011)

    Article  ADS  Google Scholar 

  35. Rastegina, E.: Number-phase uncertainty relations in terms of generalized entropies. Quant. Inf. Comput. 12, 0743 (2012)

    Google Scholar 

  36. Ballester, M.A., Wehner, S.: Entropic uncertainty relations and locking: tight bounds for mutually unbiased bases. Phys. Rev. A 75, 022319 (2007)

    Article  ADS  Google Scholar 

  37. Rastegina, A.E.: Uncertainty relations for MUBs and SIC–POVMs in terms of generalized entropies. Eur. Phys. J. D 67, 269 (2013)

    Article  ADS  Google Scholar 

  38. Rastegina, E.: Fine-grained uncertainty relations for several quantum measurements. Quantum Inf. Process. (published online)

  39. Sánchez, J.: Entropic uncertainty and certainty relations for complementary observables. Phys. Lett. A 173, 233 (1993)

    Article  ADS  Google Scholar 

  40. Ghirardi, G., Marinatto, L., Romano, R.: An optimal entropic uncertainty relation in a two-dimensional Hilbert space. Phys. Lett. A 317, 32 (2003)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  41. Srinivas, M.D.: Optimal entropic uncertainty relation for successive measurements in quantum information theory. Paramana-J. Phys. 60, 1137 (2003)

    Article  ADS  Google Scholar 

  42. Baek, K., Farrow, T., Son, W.: Optimized entropic uncertainty for successive projective measurements. Phys. Rev. A. 89, 032108 (2014)

    Article  ADS  Google Scholar 

  43. Prevedel, R., Hamel, D.R., Colbeck, R., Fisher, K., Resch, K.J.: Experimental investigation of the uncertainty principle in the presence of quantum memory and its application to witnessing entanglement. Nat. Phys. 7, 757 (2011)

    Article  Google Scholar 

  44. Li, C.-F., Xu, J.-S., Xu, X.-Y., Li, K., Guo, G.-C.: Experimental investigation of the entanglement-assisted entropic uncertainty principle. Nat. Phys. 7, 752 (2011)

    Article  Google Scholar 

  45. Tomamichel, M., Lim, C.C.W., Gisin, N., Renner, R.: Tight finite-key analysis for quantum cryptography. Nat. Commun. 3, 634 (2012)

    Article  ADS  Google Scholar 

  46. Rastegina, E.: No-cloning theorem for a single POVM. Quantum Inf. Comput. 10, 0971 (2010)

    Google Scholar 

  47. Rényi, A.: On measures of entropy and information. In: Proceedings of the Fourth Berkeley Symposium on Mathematical Statistics Probability, vol. 1, pp. 547–561. University of California Press, Berkeley (1961)

  48. Cachin, C.: Entropy measures and unconditional security in cryptography. Ph.D. dissertation, Dept. Comput. Inf. Sci., Swiss Federal Institute of Technology, Zűich, Switzerland (1997)

  49. Renner, R.: Security of quantum ket distribution. Ph.D. thesis, ETH Zurich, arXiv:0512258

Download references

Acknowledgments

This work was supported by the National Natural Science Foundation of China, under Grants numbers 11375036 and 11175033, and the Xinghai Scholar Cultivation Plan.

Author information

Authors and Affiliations

  1. School of Physics and Optoelectronic Technology, Dalian University of Technology, Dalian, 116024, China

    Jun Zhang, Yang Zhang & Chang-shui Yu

Authors
  1. Jun Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yang Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Chang-shui Yu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Chang-shui Yu.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Zhang, J., Zhang, Y. & Yu, Cs. Rényi entropy uncertainty relation for successive projective measurements. Quantum Inf Process 14, 2239–2253 (2015). https://doi.org/10.1007/s11128-015-0950-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11128-015-0950-z

Keywords

Search

Navigation