Skip to main content
SpringerLink
Log in

A Dynamical Uncertainty Principle in von Neumann Algebras by Operator Monotone Functions

  • Published:
Journal of Statistical Physics Aims and scope Submit manuscript

Abstract

Suppose that A 1,…,A N are observables (selfadjoint matrices) and ρ is a state (density matrix). In this case the standard uncertainty principle, proved by Robertson, gives a bound for the quantum generalized variance, namely for det {Cov  ρ (A j ,A k )}, using the commutators [A j ,A k ]; this bound is trivial when N is odd. Recently a different inequality of Robertson-type has been proved by the authors with the help of the theory of operator monotone functions. In this case the bound makes use of the commutators [ρ,A j ] and is non-trivial for any N. In the present paper we generalize this new result to the von Neumann algebra case. Nevertheless the proof appears to simplify all the existing ones.

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. Andai, A.: Uncertainty principle with quantum Fisher information. J. Math. Phys. 49, 012106 (2008)

    Article  MathSciNet  ADS  Google Scholar 

  2. Bhatia, R.: Matrix Analysis. Springer, New York (1996)

    MATH  Google Scholar 

  3. Daoud, M.: Representations and properties of generalized A r statistics, coherent states and Robertson uncertainty relations. J. Phys. A Math. Gen. 39, 889–901 (2006)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  4. Dodonov, A.V., Dodonov, V.V., Mizrahi, S.S.: Separability dynamics of two-mode Gaussian states in parametric conversion and amplification. J. Phys. A Math. Gen. 38, 683–696 (2005)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  5. Gibilisco, P., Isola, T.: Uncertainty principle and quantum Fisher information. Ann. Inst. Stat. Math 59, 147–159 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  6. Gibilisco, P., Isola, T.: Uncertainty principle for Wigner-Yanase-Dyson information in semifinite von Neumann algebras. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 11(1), 127–133 (2008)

    Article  MathSciNet  Google Scholar 

  7. Gibilisco, P., Isola, T.: An inequality related to uncertainty principle in von Neumann algebras. Int. J. Math. (2008, to appear). arXiv:0804.2651v1

  8. Gibilisco, P., Hiai, F., Petz, D.: Quantum covariance, quantum Fisher information and the uncertainty principle. arXiv:0712.1208v1 (2007)

  9. Gibilisco, P., Imparato, D., Isola, T.: Uncertainty principle and quantum Fisher information, II. J. Math. Phys. 48, 072109 (2007)

    Article  MathSciNet  ADS  Google Scholar 

  10. Gibilisco, P., Imparato, D., Isola, T.: A volume inequality for quantum Fisher information and the uncertainty principle. J. Stat. Phys. 130(3), 545–559 (2008)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  11. Gibilisco, P., Imparato, D., Isola, T.: A Robertson-type uncertainty principle and quantum Fisher information. Linear Algebra Appl. 428, 1706–1724 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  12. Hansen, F.: Metric adjusted skew information. arXiv:math-ph/0607049v3 (2006)

  13. Jarvis, P.D., Morgan, S.O.: Born reciprocity and the granularity of spacetime. Found. Phys. Lett. 19, 501 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  14. Kato, T.: Perturbation Theory for Linear Operators. Springer, New York (1966)

    MATH  Google Scholar 

  15. Kosaki, H.: Matrix trace inequality related to uncertainty principle. Int. J. Math. 16, 629–645 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  16. Luo, S.: Quantum Fisher information and uncertainty relations. Lett. Math. Phys. 53, 243–251 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  17. Luo, S.: Wigner-Yanase skew information and uncertainty relations. Phys. Rev. Lett. 91, 180403 (2003)

    Article  ADS  Google Scholar 

  18. Luo, S., Zhang, Q.: On skew information. IEEE Trans. Inf. Theory 50, 1778–1782 (2004)

    Article  MathSciNet  Google Scholar 

  19. Luo, S., Zhang, Z.: An informational characterization of Schrödinger’s uncertainty relations. J. Stat. Phys. 114, 1557–1576 (2004)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  20. Luo, S., Zhang, Q.: Correction to “On skew information”. IEEE Trans. Inf. Theory 51, 4432 (2005)

    Article  MathSciNet  Google Scholar 

  21. Petz, D.: Monotone Metrics on Matrix Spaces. Linear Algebra Appl. 244, 81–96 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  22. Petz, D., Hasegawa, H.: On the Riemannian metric of α-entropies of density matrices. Lett. Math. Phys. 38, 221–225 (1996)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  23. Robertson, H.P.: An indeterminacy relation for several observables and its classical interpretation. Phys. Rev. 46, 794–801 (1934)

    Article  MATH  ADS  Google Scholar 

  24. Takesaki, M.: Theory of Operator Algebras, vol. I. Encyclopaedia of Mathematical Sciences, vol. 124. Springer, Berlin (2002)

    Google Scholar 

  25. Takesaki, M.: Theory of Operator Algebras, vol. II. Encyclopaedia of Mathematical Sciences, vol. 125. Springer, Berlin (2002)

    Google Scholar 

  26. Takesaki, M.: Theory of Operator Algebras, vol. III. Encyclopaedia of Mathematical Sciences, vol. 127. Springer, Berlin (2003)

    Google Scholar 

  27. Trifonov, D.A.: Generalized intelligent states and squeezing. J. Math. Phys. 35, 2297–2308 (1994)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  28. Trifonov, D.A.: State extended uncertainty relations. J. Phys. A Math. Gen. 33, 299–304 (2000)

    Article  MathSciNet  ADS  Google Scholar 

  29. Trifonov, D.A.: Generalizations of Heisenberg uncertainty relation. Eur. Phys. J. B 29, 349–353 (2002)

    Article  ADS  Google Scholar 

  30. Yanagi, K., Furuichi, S., Kuriyama, K.: A generalized skew information and uncertainty relation. IEEE Trans. Inf. Theory 51, 4401–4404 (2005)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Dipartimento SEFEMEQ, Facoltà di Economia, Università di Roma “Tor Vergata”, Via Columbia 2, 00133, Rome, Italy

    Paolo Gibilisco

  2. Dipartimento di Matematica, Università di Roma “Tor Vergata”, Via della Ricerca Scientifica, 00133, Rome, Italy

    Tommaso Isola

Authors
  1. Paolo Gibilisco
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Tommaso Isola
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Paolo Gibilisco.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Gibilisco, P., Isola, T. A Dynamical Uncertainty Principle in von Neumann Algebras by Operator Monotone Functions. J Stat Phys 132, 937–944 (2008). https://doi.org/10.1007/s10955-008-9582-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10955-008-9582-3

Keywords

Search

Navigation