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Uniform Continuity of POVMs

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Abstract

Recently a characterization of uniformly continuous POVMs and a necessary condition for a uniformly continuous POVM F to have the norm-1 property have been provided. Moreover it was proved that in the commutative case, uniform continuity corresponds to the existence of a Feller Markov kernel. We apply such results to the analysis of some relevant physical examples; i.e., the phase space localization observables, the unsharp phase observable and the unsharp number observable of which we study the uniform continuity, the norm-1 property and the existence of a Feller Markov kernel.

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Notes

  1. Since

    $$ \lim_{m\to\infty} \begin{pmatrix} m\\ n \end{pmatrix} \epsilon^n (1-\epsilon)^{m-n}=\lim _{m\to\infty}\frac{\epsilon^n}{n!} \bigl[m(1-\epsilon)^{m-n/n} \bigr]\cdots\bigl[(m-n) (1-\epsilon)^{m-n/n} \bigr]=0,$$

    Notice that \(F_{n}^{\epsilon}\) is compact for every \(n\in\mathbb{N}\) (see pp. 234–235 in [33]).

References

  1. Ali, S.T.: A geometrical property of POV-measures and systems of covariance. In: Doebner, H.-D., Andersson, S.I., Petry, H.R. (eds.) Differential Geometric Methods in Mathematical Physics. Lecture Notes in Mathematics, vol. 905, pp. 207–228. Springer, Berlin (1982)

    Chapter  Google Scholar 

  2. Ali, S.T., Emch, G.G.: J. Math. Phys. 15, 176 (1974)

    Article  ADS  MathSciNet  Google Scholar 

  3. Ali, S.T., Prugovečki, E.: Physica A 89, 501–521 (1977)

    Article  ADS  MathSciNet  Google Scholar 

  4. Ali, S.T., Carmeli, C., Heinosaari, T., Toigo, A.: Found. Phys. 39, 593–612 (2009)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  5. Beneduci, R.: Semispectral measures and Feller Markov kernels (2012). arXiv:1207.0086v1

  6. Beneduci, R., Schroeck, F. Jr.: A note on the relationship between localization and the norm-1 property. J. Phys. A, Math. Theor. 46, 305303 (2013)

    Article  MathSciNet  Google Scholar 

  7. Beneduci, R., Nisticó, G.: J. Math. Phys. 44, 5461 (2003)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  8. Beneduci, R.: J. Math. Phys. 47, 062104 (2006)

    Article  ADS  MathSciNet  Google Scholar 

  9. Beneduci, R.: Int. J. Geom. Methods Mod. Phys. 3, 1559 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  10. Beneduci, R.: J. Math. Phys. 48, 022102 (2007)

    Article  ADS  MathSciNet  Google Scholar 

  11. Beneduci, R.: Il Nuovo Cimento B 123, 43–62 (2008)

    ADS  MathSciNet  Google Scholar 

  12. Beneduci, R.: Int. J. Theor. Phys. 49, 3030–3038 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  13. Beneduci, R.: Bull. Lond. Math. Soc. 42, 441–451 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  14. Beneduci, R.: Linear Algebra Appl. 433, 1224–1239 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  15. Beneduci, R.: Int. J. Theor. Phys. 49, 3030–3038 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  16. Beneduci, R.: Int. J. Theor. Phys. 50, 3724–3736 (2011). doi:10.1007/s10773-011-0907-7

    Article  MATH  MathSciNet  Google Scholar 

  17. Beneduci, R., Brooke, J., Curran, R., Schroeck, F. Jr.: Int. J. Theor. Phys. 50, 3682–3696 (2011). doi:10.1007/s10773-011-0797-8

    Article  MATH  MathSciNet  Google Scholar 

  18. Beneduci, R., Brooke, J., Curran, R., Schroeck, F. Jr.: Int. J. Theor. Phys. 50, 3697–3723 (2011). doi:10.1007/s10773-011-0869-9

    Article  MATH  MathSciNet  Google Scholar 

  19. Berberian, S.K.: Notes on Spectral Theory. Van Nostrand Mathematical Studies. Van Nostrand, New York (1966)

    MATH  Google Scholar 

  20. Busch, P., Grabowski, M., Lahti, P.: Operational Quantum Physics. Lecture Notes in Physics, vol. 31. Springer, Berlin (1995)

    MATH  Google Scholar 

  21. Davies, E.B., Lewis, J.T.: Commun. Math. Phys. 17, 239 (1970)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  22. Heinoinen, T., Lahti, P., Pellonpää, J.P., Pulmannová, S., Ylinen, K.: J. Math. Phys. 44, 1998–2008 (2003)

    Article  ADS  MathSciNet  Google Scholar 

  23. Holevo, A.S.: Probabilistics and Statistical Aspects of Quantum Theory. North-Holland, Amsterdam (1982)

    Google Scholar 

  24. Holevo, A.S.: Trans. Mosc. Math. Soc. 26, 133 (1972)

    MATH  MathSciNet  Google Scholar 

  25. Jenčová, A., Pulmannová, S.: Rep. Math. Phys. 59, 257–266 (2007)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  26. Jenčová, A., Pulmannová, A.: Found. Phys. 39, 613–624 (2009)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  27. Kraus, K.: Position observables of the photon. In: Price, W.C., Chissick, S.S. (eds.) The Uncertainty Principle and Foundations of Quantum Mechanics, pp. 293–320. Wiley, London (1977)

    Google Scholar 

  28. Munkres, J.R.: Topology. Prentice Hall, Upper Saddle River (2000)

    MATH  Google Scholar 

  29. Naimark, M.A.: Izv. Akad. Nauk SSSR, Ser. Mat. 4, 277–318 (1940)

    MATH  Google Scholar 

  30. Newton, T.D., Wigner, E.P.: Rev. Mod. Phys. 21, 400 (1949)

    Article  ADS  MATH  Google Scholar 

  31. Prugovečki, E.: Stochastic Quantum Mechanics and Quantum Spacetime. Reidel, Dordrecht (1984)

    Book  MATH  Google Scholar 

  32. Reed, M., Simon, B.: Methods of Modern Mathematical Physics. Academic Press, New York (1980)

    MATH  Google Scholar 

  33. Riesz, F., Nagy, B.S.: Functional Analysis. Dover, New York (1990)

    MATH  Google Scholar 

  34. Schroeck, F.E. Jr.: Quantum Mechanics on Phase Space. Kluwer Academic Publishers, Dordrecht (1996)

    Book  MATH  Google Scholar 

  35. Wightman, A.S.: Rev. Mod. Phys. 34, 845–872 (1962)

    Article  ADS  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Physics, University of Calabria, Calabria, Italy

    Roberto Beneduci

  2. INFN Gruppo Collegato Cosenza, via P. Bucci cubo 30-B, 87036, Arcavacata di Rende, Italy

    Roberto Beneduci

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  1. Roberto Beneduci
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Correspondence to Roberto Beneduci.

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Beneduci, R. Uniform Continuity of POVMs. Int J Theor Phys 53, 3531–3545 (2014). https://doi.org/10.1007/s10773-013-1883-x

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