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Shannon entropy as a measure of uncertainty in positions and momenta

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Journal of Russian Laser Research Aims and scope

Abstract

I start with a brief report of the topic of entropic uncertainty relations for the position and momentum variables. Then I investigate the discrete Shannon entropies related to the case of a finite number of detectors set to measure the probability distributions in the position and momentum spaces. I derive the uncertainty relation for the sum of the Shannon entropies which generalizes the previous approach by I. Bialynicki-Birula based on an infinite number of detectors (bins).

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References

  1. I. Bialynicki-Birula and J. Mycielski, Commun. Math. Phys., 44, 129 (1975).

    Article  MathSciNet  ADS  Google Scholar 

  2. D. Deutsch, Phys. Rev. Lett., 50, 631 (1983).

    Article  MathSciNet  ADS  Google Scholar 

  3. H. Maassen and J. B. M. Uffink, Phys. Rev. Lett., 60, 1103 (1988).

    Article  MathSciNet  ADS  Google Scholar 

  4. M. H. Partovi, Phys. Rev. Lett., 50, 1883 (1983).

    Article  MathSciNet  ADS  Google Scholar 

  5. I. Bialynicki-Birula, Phys. Lett. A, 103, 253 (1984).

    Article  MathSciNet  ADS  Google Scholar 

  6. I. Bialynicki-Birula, Phys. Rev. A, 74, 052101 (2006).

    Article  MathSciNet  ADS  Google Scholar 

  7. M. D. Srinivas, Pramana, 25, 369 (1985).

    Article  ADS  Google Scholar 

  8. I. Bialynicki-Birula and Ł. Rudnicki, “Entropic uncertainty relations in quantum physics,” in: K. D. Sen (ed.), Statistical Complexity: Applications in Electronic Structure, Springer (2011) [arXiv:1001.4668 (2010)].

  9. V. V. Dodonov, J. Opt. B: Quantum Semiclass. Opt., 4, 98 (2002).

    Article  MathSciNet  ADS  Google Scholar 

  10. B. Coutinho dos Santos, K. Dechoum, and A. Z. Khoury, Phys. Rev. Lett., 103, 230503 (2009).

    Article  Google Scholar 

  11. A. Saboia, F. Toscano, and S. P. Walborn, Phys. Rev. A, 83, 032307 (2011).

    Article  ADS  Google Scholar 

  12. A. Peres, Quantum Theory: Concepts and Methods, Kluwer, Dordrecht (1995).

    MATH  Google Scholar 

  13. D. Chafaї, “Gaussian maximum of entropy and reversed log-Sobolev inequality,” Séminaire de probabilitiés, Strasbourg (2002), Vol. 36, p. 194.

  14. G. Wilk and Z. Włodarczyk, Phys. Rev. A, 79, 062108 (2009).

    Article  MathSciNet  ADS  Google Scholar 

  15. I. Bialynicki-Birula and Ł. Rudnicki, Phys. Rev. A, 81, 026101 (2010).

    Article  ADS  Google Scholar 

  16. Ł. Rudnicki, “Uncertainty related to position and momentum localization of a quantum state,” arXiv:1010.3269v1 (2010).

  17. Ł. Rudnicki, S. P. Walborn, and F. Toscano (in preparation).

  18. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, Dover, New York (1964).

    MATH  Google Scholar 

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  1. Center for Theoretical Physics, Polish Academy of Sciences Aleja Lotników 32/46, PL-02-668, Warsaw, Poland

    Łukasz Rudnicki

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  1. Łukasz Rudnicki
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Correspondence to Łukasz Rudnicki.

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Rudnicki, Ł. Shannon entropy as a measure of uncertainty in positions and momenta. J Russ Laser Res 32, 393–399 (2011). https://doi.org/10.1007/s10946-011-9227-x

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  • DOI: https://doi.org/10.1007/s10946-011-9227-x

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