Skip to main content
SpringerLink
Log in

Compatibility Properties of Extreme Quantum Observables

  • Published:
Letters in Mathematical Physics Aims and scope Submit manuscript

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.

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. Arveson W.: Subalgebras of C*-algebras. Acta Math. 123, 141–224 (1969)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  3. Heinosaari T., Reitzner D., Stano P.: Notes on joint measurability of quantum observables. Found. Phys. 38, 1133–1147 (2008)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  4. Jenčová A., Pulmannová S.: How sharp are PV measures. Rep. Math. Phys. 59, 257 (2007)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  5. Jenčová A., Pulmannová S.: Characterizations of commutative POV measures. Found. Phys. 39, 613 (2009)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  6. Jenčová A., Pulmannová S., Vinceková E.: Sharp and fuzzy observables on effect algebras. Int. J. Theor. Phys. 47, 125 (2008)

    Article  MATH  Google Scholar 

  7. Lahti P.: Coexistence and joint measurability in quantum mechanics. Int. J. Theor. Phys. 42, 893 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  8. Lahti P., Pulmannová S.: Coexistent observables and effects in quantum mechanics. Rep. Math. Phys. 39, 339 (1997)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  9. Lahti P., Ylinen K.: Dilations of positive operator measures and bimeasures related to quantum mechanics. Math. Slovaca 54, 169 (2004)

    MATH  MathSciNet  Google Scholar 

  10. Ludwig, G.: Foundations of quantum mechanics, vol. I, Springer, Berlin (1983)

  11. Pellonpää J.-P.: Complete characterization of extreme quantum observables in infinite dimensions. J. Phys. A Math. Theor. 44, 085304 (2011)

    Article  ADS  Google Scholar 

  12. Pellonpää J.-P.: “Quantum instruments: I. Extreme instruments. J. Phys. A Math. Theor. 46, 025302 (2013)

    Article  ADS  Google Scholar 

  13. Pellonpää J.-P.: On coexistence and joint measurability of rank-1 quantum observables. J. Phys. A Math. Theor. 47, 052002 (2014)

    Article  ADS  Google Scholar 

  14. Quintino M., Vértesi T., Brunner N.: Joint measurability, einstein-podolsky-rosen steering, and bell nonlocality. Phys. Rev. Lett. 113, 160402 (2014)

    Article  ADS  Google Scholar 

  15. Reeb D., Reitzner D., Wolf M.M.: Coexistence does not imply joint measurability. J. Phys. A Math. Theor. 46, 462002 (2013)

    Article  ADS  MathSciNet  Google Scholar 

  16. Uola R., Moroder T., Gühne O.: Joint measurability of generalised measurements implies classicality. Phys. Rev. Lett. 113, 160403 (2014)

    Article  ADS  Google Scholar 

  17. Ylinen K.: Positive operator bimeasures and a noncommutative generalization. Studia Math. 118, 157 (1996)

    MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Physics and Astronomy, Turku Centre for Quantum Physics, University of Turku, 20014, Turku, Finland

    Erkka Haapasalo, Juha-Pekka Pellonpää & Roope Uola

  2. Naturwissenschaftlich-Technische Fakultät, Universität Siegen, Walter-Flex-Str. 3, 57068, Siegen, Germany

    Roope Uola

Authors
  1. Erkka Haapasalo
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Juha-Pekka Pellonpää
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Roope Uola
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Erkka Haapasalo.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Haapasalo, E., Pellonpää, JP. & Uola, R. Compatibility Properties of Extreme Quantum Observables. Lett Math Phys 105, 661–673 (2015). https://doi.org/10.1007/s11005-015-0754-1

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11005-015-0754-1

Mathematics Subject Classification

Keywords

Search

Navigation