Universal Randomization of Quantum Observables

  • Roberto BeneduciEmail author


Commutative POVMs are smearing of spectral measures with the smearing realized by a Markov kernel. In the case of POVMs defined on the Borel σ-algebra of the reals, the existence of a universal Markov kernel has been established. Here we show that every weak Markov kernel is functionally subordinated to the universal Markov kernel. Then, we analyze the implications of that result to the characterization of a Markov kernel as a Choquet’s integral on the space of deterministic Markov kernels.


Positive operator valued measures Markov kernels von Neumann algebras Choquet’s integral Quantum mechanics Joint measurability 



The present work was performed under the auspices of the GNFM (Gruppo Nazionale di Fisica Matematica).


  1. 1.
    Ali, S.T., Emch, G.G.: Fuzzy observables in quantum mechanics. J. Math. Phys. 15, 176 (1974)ADSMathSciNetCrossRefGoogle Scholar
  2. 2.
    Ali, S.T.: A geometrical property of POV-measures and systems of covariance. Lect. Notes Math. 905, 207–228 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Beneduci, R.: Commutative POVM-measures: from the Choquet representation to the Markov kernel and back. Russ. J. Math. Phys. 25, 158–182 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Beneduci, R.: Universal Markov kernels for quantum observables, Geometric Methods in Physics. XXXVI Workshop 2017, Trends in Mathematics, 21–29 (2018)Google Scholar
  5. 5.
    Beneduci, R., Nisticó, G.: Sharp reconstruction of unsharp quantum observables. J. Math. Phys. 44, 5461 (2003)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Beneduci, R.: A geometrical characterizations of commutative positive operator valued measures. J. Math. Phys. 47, 062104 (2006)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Beneduci, R.: Joint measurability through Naimark’s dilation theorem. Rep. Math. Phys. 79, 197–213 (2017)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Beneduci, R.: Positive operator valued measures and feller Markov kernels. J. Math. Anal. Appl. 442, 50–71 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Beneduci, R.: Unsharp number observable and Neumark theorem. Il Nuovo Cimento B 123, 43 (2008)ADSMathSciNetGoogle Scholar
  10. 10.
    Beneduci, R.: Unsharpness, Naimark theorem and informational equivalence of quantum observables. Int. J. Theor. Phys. 49, 3030 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Beneduci, R.: Infinite sequences of linear functionals, positive operator-valued measures and Naimark extension theorem. Bull. Lond. Math. Soc. 42, 441–451 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Beneduci, R.: Stochastic matrices and a property of the infinite sequences of linear functionals. Linear Algebra Appl. 43, 1224–1239 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Beneduci, R.: On the relationships between the moments of a POVM and the generator of the von Neumann algebra it generates. Int. J. Theor. Phys. 50, 3724–3736 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Beneduci, R., Brooke, J., Curran, R., Schroeck, F.E.: Classical mechanics in Hilbert space, part I. Int. J. Theor. Phys. 50, 3682–3696 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Beneduci, R., Brooke, J., Curran, R., Schroeck, F.: Classical mechanics in Hilbert space, part II. Int. J. Theor. Phys. 50, 3697–3723 (2011)CrossRefzbMATHGoogle Scholar
  16. 16.
    Busch, P., Grabowski, M., Lahti, P.: Operational quantum physics. Lect. Notes Phys. 31 (1995)Google Scholar
  17. 17.
    Busch, P., Lahti, P., Werner, R.: Proof of Heisenberg’s error-disturbance relation. Phys. Rev. Lett. 111, 160405 (2013)ADSCrossRefGoogle Scholar
  18. 18.
    Davies, E.B.: Quantum Mechanics of Open Systems. Academic Press, London (1976)zbMATHGoogle Scholar
  19. 19.
    Gudder, S.P.: Stochastic Methods in Quantum Mechanics. North Holland, New York (2005)Google Scholar
  20. 20.
    Holevo, A.S.: Probabilistics and Statistical Aspects of Quantum Theory. North Holland, Amsterdam (1982)Google Scholar
  21. 21.
    Jenčová, A., Pulmannová, S.: Rep. Math. Phys. 59, 257–266 (2007)ADSMathSciNetCrossRefGoogle Scholar
  22. 22.
    Jenčovà, A., Pulmannovà, S.: Characterizations of commutative POV measures. Found. Phys. 39, 613–624 (2009)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Lahti, P.: Coexistence and joint measurability in quantum mechanics. Int. J. Theor. Phys. 42, 893–906 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Munroe, M.E.: Introduction to Measure and Integration. Addison-Wesley Publishing Company, Reading (1953)zbMATHGoogle Scholar
  25. 25.
    Naimark, M.A., Akad, I.: Nauk SSSR Ser. Mat. 4, 277–318 (1940)Google Scholar
  26. 26.
    Prugovečki, E.: Stochastic Quantum Mechanics and Quantum Spacetime. Reidel Publishing Company, Dordrecht (1984)CrossRefzbMATHGoogle Scholar
  27. 27.
    Riesz, F., Nagy, B.S.: Functional Analysis, Dover (1990)Google Scholar
  28. 28.
    Schroeck, F.E. Jr.: Quantum Mechanics on Phase Space. Kluwer Academic Publishers, Dordrecht (1996)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of PhysicsUniversity of Calabria and INFN gruppo collegato Cosenza via P. Bucci cubo 31-CArcavacata di RendeItaly

Personalised recommendations