Advertisement

Letters in Mathematical Physics

, Volume 105, Issue 5, pp 661–673 | Cite as

Compatibility Properties of Extreme Quantum Observables

  • Erkka HaapasaloEmail author
  • Juha-Pekka Pellonpää
  • Roope Uola
Article

Abstract

Recently, a problem concerning the equivalence of joint measurability and coexistence of quantum observables was solved (Reeb at al. J Phys A Math Theor 46:462002, 2013). In this paper, we generalize two known joint measurability results from sharp observables to the class of extreme observables and study relationships between coexistence, joint measurability, and post-processing of quantum observables when an extreme observable is involved. We also discuss another notion of compatibility and provide a counterexample separating this from the former notions.

Keywords

positive operator-valued measure joint measurability coexistence extremality 

Mathematics Subject Classification

81P45 81Q99 46N10 46N50 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Arveson W.: Subalgebras of C*-algebras. Acta Math. 123, 141–224 (1969)CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    Haapasalo E., Heinosaari T., Pellonpää J.-P.: When do pieces determine the whole? Extremal marginals of a completely positive map. Rev. Math. Phys. 26, 1450002 (2014)CrossRefMathSciNetGoogle Scholar
  3. 3.
    Heinosaari T., Reitzner D., Stano P.: Notes on joint measurability of quantum observables. Found. Phys. 38, 1133–1147 (2008)CrossRefADSzbMATHMathSciNetGoogle Scholar
  4. 4.
    Jenčová A., Pulmannová S.: How sharp are PV measures. Rep. Math. Phys. 59, 257 (2007)CrossRefADSzbMATHMathSciNetGoogle Scholar
  5. 5.
    Jenčová A., Pulmannová S.: Characterizations of commutative POV measures. Found. Phys. 39, 613 (2009)CrossRefADSzbMATHMathSciNetGoogle Scholar
  6. 6.
    Jenčová A., Pulmannová S., Vinceková E.: Sharp and fuzzy observables on effect algebras. Int. J. Theor. Phys. 47, 125 (2008)CrossRefzbMATHGoogle Scholar
  7. 7.
    Lahti P.: Coexistence and joint measurability in quantum mechanics. Int. J. Theor. Phys. 42, 893 (2003)CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Lahti P., Pulmannová S.: Coexistent observables and effects in quantum mechanics. Rep. Math. Phys. 39, 339 (1997)CrossRefADSzbMATHMathSciNetGoogle Scholar
  9. 9.
    Lahti P., Ylinen K.: Dilations of positive operator measures and bimeasures related to quantum mechanics. Math. Slovaca 54, 169 (2004)zbMATHMathSciNetGoogle Scholar
  10. 10.
    Ludwig, G.: Foundations of quantum mechanics, vol. I, Springer, Berlin (1983)Google Scholar
  11. 11.
    Pellonpää J.-P.: Complete characterization of extreme quantum observables in infinite dimensions. J. Phys. A Math. Theor. 44, 085304 (2011)CrossRefADSGoogle Scholar
  12. 12.
    Pellonpää J.-P.: “Quantum instruments: I. Extreme instruments. J. Phys. A Math. Theor. 46, 025302 (2013)CrossRefADSGoogle Scholar
  13. 13.
    Pellonpää J.-P.: On coexistence and joint measurability of rank-1 quantum observables. J. Phys. A Math. Theor. 47, 052002 (2014)CrossRefADSGoogle Scholar
  14. 14.
    Quintino M., Vértesi T., Brunner N.: Joint measurability, einstein-podolsky-rosen steering, and bell nonlocality. Phys. Rev. Lett. 113, 160402 (2014)CrossRefADSGoogle Scholar
  15. 15.
    Reeb D., Reitzner D., Wolf M.M.: Coexistence does not imply joint measurability. J. Phys. A Math. Theor. 46, 462002 (2013)CrossRefADSMathSciNetGoogle Scholar
  16. 16.
    Uola R., Moroder T., Gühne O.: Joint measurability of generalised measurements implies classicality. Phys. Rev. Lett. 113, 160403 (2014)CrossRefADSGoogle Scholar
  17. 17.
    Ylinen K.: Positive operator bimeasures and a noncommutative generalization. Studia Math. 118, 157 (1996)zbMATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2015

Authors and Affiliations

  • Erkka Haapasalo
    • 1
    Email author
  • Juha-Pekka Pellonpää
    • 1
  • Roope Uola
    • 1
    • 2
  1. 1.Department of Physics and Astronomy, Turku Centre for Quantum PhysicsUniversity of TurkuTurkuFinland
  2. 2.Naturwissenschaftlich-Technische FakultätUniversität SiegenSiegenGermany

Personalised recommendations