Skip to main content
SpringerLink
Log in

Non-commutative Rényi entropic uncertainty principles

  • Articles
  • Published:
Science China Mathematics Aims and scope Submit manuscript

Abstract

In this paper, we calculate the norm of the string Fourier transform on subfactor planar algebras and characterize the extremizers of the inequalities for parameters 0 < p,q ⩽ ∞. Furthermore, we establish Rényi entropic uncertainty principles for subfactor planar algebras.

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. Babenko K I. An inequality in the theory of Fourier integrals. Izv Akad Nauk SSSR Ser Mat, 1961, 25: 531–542

    MathSciNet  MATH  Google Scholar 

  2. Beck C, Graudenz D. Symbolic dynamics of successive quantum-mechanical measurements. Phys Rev A (3), 1992, 46: 6265–6276

    Article  MathSciNet  Google Scholar 

  3. Beckner W. Inequalities in Fourier analysis. Ann of Math (2), 1975, 102: 159–182

    Article  MathSciNet  Google Scholar 

  4. Bialynicki-Birula I. Formulation of the uncertainty relations in terms of the Renyi entropies. Phys Rev A (3), 2006, 74: 052101

    Article  MathSciNet  Google Scholar 

  5. Bovino F, Castagnoli G, Ekert A, et al. Direct measurement of nonlinear properties of bipartite quantum states. Phys Rev Lett, 2005, 95: 240407

    Article  MathSciNet  Google Scholar 

  6. Evans D, Kawahigashi Y. Quantum Symmetries on Operator Algebras. Oxford: Clarendon Press, 1998

    MATH  Google Scholar 

  7. Gilbert J, Rzeszotnik Z. The norm of the Fourier transform on finite abelian groups. Ann Inst Fourier Grenoble, 2010, 60: 1317–1346

    Article  MathSciNet  Google Scholar 

  8. Giovannetti V, Lloyd S. Additivity properties of a Gaussian channel. Phys Rev A (3), 2004, 69: 062307

    Article  MathSciNet  Google Scholar 

  9. Gühne O, Lewenstein M. Entropic uncertainty relations and entanglement. Phys Rev A (3), 2004, 70: 022316

    Article  MathSciNet  Google Scholar 

  10. Hardy G H, Littlewood J E. Some new properties of Fourier constants. Math Ann, 1927, 97: 159–209

    Article  MathSciNet  Google Scholar 

  11. Hewitt E, Hirschman I. A maximum problem in harmonic analysis. Amer J Math, 1954, 76: 839–852

    Article  MathSciNet  Google Scholar 

  12. Hirschman I I. A note on entropy. Amer J Math, 1957, 79: 152–156

    Article  MathSciNet  Google Scholar 

  13. Jaffe A, Liu Z. Planar para algebras, reflection positivity. Comm Math Phys, 2017, 352: 95–133

    Article  MathSciNet  Google Scholar 

  14. Jiang C, Liu Z, Wu J. Noncommutative uncertainty principles. J Funct Anal, 2016, 270: 264–311

    Article  MathSciNet  Google Scholar 

  15. Jiang C, Liu Z, Wu J. Uncertainty principles for locally compact quantum groups. J Funct Anal, 2018, 274: 2399–2445

    Article  MathSciNet  Google Scholar 

  16. Jiang C L, Liu Z W, Wu J S. Block maps and Fourier analysis. Sci China Math, 2019, 62: 1585–1614

    Article  MathSciNet  Google Scholar 

  17. Jones V. Index for subfactors. Invent Math, 1983, 72: 1–25

    Article  MathSciNet  Google Scholar 

  18. Jones V. Planar algebra, I. ArXiv:math/9909027, 1999

  19. Kodiyalam V, Landau Z, Sunder V. The planar algebra associated to a Kac algebra. Proc Indian Acad Sci Math Sci, 2003, 113: 15–51

    Article  MathSciNet  Google Scholar 

  20. Kustermans J, Vaes S. Locally compact quantum groups. Ann Sci Éc Norm Supér (4), 2000, 33: 837–934

    Article  MathSciNet  Google Scholar 

  21. Levay P, Nagy S, Pipek J. Elementary formula for entanglement entropies of fermionic systems. Phys Rev A (3), 2005, 72: 022302

    Article  MathSciNet  Google Scholar 

  22. Liu Z. Exchange relation planar algebras of small rank. Trans Amer Math Soc, 2016, 308: 8303–8348

    Article  MathSciNet  Google Scholar 

  23. Liu Z, Wu J. Noncommutative Fourier transform: A survey (in Chinese). Acta Math Sinica Chin Ser, 2017, 60: 81–96

    MathSciNet  Google Scholar 

  24. Murray F, von Neumann J. On rings of operators. Ann of Math (2), 1936, 37: 116–229

    Article  MathSciNet  Google Scholar 

  25. Ocneanu A. Quantised groups, string algebras and Galois theory for algebras. In: Operator Algebras and Applications, vol. 2. London Mathematical Society Lecture Note Series, vol. 136. Cambridge: Cambridge University Press, 1988, 119–172

    Google Scholar 

  26. Pimsner M, Popa S. Entropy and index for subfactors. Ann Sci Éc Norm Supér (4), 1986, 19: 57–106

    Article  MathSciNet  Google Scholar 

  27. Pisier G, Xu Q. Non-commutative Lp-spaces. In: Handbook of the Geometry of Banach Spaces, vol. 2. Amsterdam: North-Holland, 2003, 1459–1517

    Chapter  Google Scholar 

  28. Renner R, Gisin N, Kraus B. Information-theoretic security proof for quantum-key-distribution protocols. Phys Rev A (3), 2005, 72: 012332

    Article  Google Scholar 

  29. Rényi A. On measures of information and entropy. In: Proceedings of the 4th Berkeley Symposium on Mathematics, Statistics and Probability. Berkeley: University of California Press, 1960, 547–561

    Google Scholar 

  30. Russo B. The norm of the Lp-Fourier transform on unimodular groups. Trans Amer Math Soc, 1974, 192: 293–305

    MathSciNet  MATH  Google Scholar 

  31. Tsallis C. Possible generalization of Boltzmann-Gibbs statistics. J Stat Phys, 1988, 52: 479–487.

    Article  MathSciNet  Google Scholar 

  32. Wilk G, Wlodarczyk Z. Uncertainty relations in terms of the Tsallis entropy. Phys Rev A (3), 2009, 79: 062108

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The first author was supported by Templeton Religion Trust (Grant No. TRT 0159). The second author was supported by National Natural Science Foundation of China (Grant No. 11771413) and Templeton Religion Trust (Grant No. TRT 0159). Part of the work was done during visits of Zhengwei Liu and Jinsong Wu to Hebei Normal University and of Jinsong Wu to Harvard University. The authors thank the referees for careful reading.

Author information

Authors and Affiliations

  1. Department of Mathematics and Department of Physics, Harvard University, Cambridge, MA, 02138, USA

    Zhengwei Liu

  2. Institute of Advanced Study in Mathematics, Harbin Institute of Technology, Harbin, 150001, China

    Jinsong Wu

Authors
  1. Zhengwei Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jinsong Wu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Zhengwei Liu.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Liu, Z., Wu, J. Non-commutative Rényi entropic uncertainty principles. Sci. China Math. 63, 2287–2298 (2020). https://doi.org/10.1007/s11425-019-9523-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11425-019-9523-4

Keywords

MSC(2010)

Search

Navigation