Skip to main content
SpringerLink
Log in

Strongly Incompatible Quantum Devices

Foundations of Physics Aims and scope Submit manuscript

Cite this article

Abstract

The fact that there are quantum observables without a simultaneous measurement is one of the fundamental characteristics of quantum mechanics. In this work we expand the concept of joint measurability to all kinds of possible measurement devices, and we call this relation compatibility. Two devices are incompatible if they cannot be implemented as parts of a single measurement setup. We introduce also a more stringent notion of incompatibility, strong incompatibility. Both incompatibility and strong incompatibility are rigorously characterized and their difference is demonstrated by examples.

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

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5

Notes

  1. This set is equipped with the natural σ-algebra \(\mathcal{F}=2^{\varOmega}\) containing all subsets of Ω. Thus X⊆Ω is equivalent to \(X\in\mathcal{F}\) in this paper, while the latter should be employed in treating infinite outcome set Ω.

References

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

    Article  MATH  MathSciNet  Google Scholar 

  2. Busch, P.: “No Information Without Disturbance”: quantum limitations of measurement. In: Christian, J., Myrvold, W. (eds.) Quantum Reality, Relativistic Causality, and Closing the Epistemic Circle. Springer, Berlin (2009)

    Google Scholar 

  3. Busch, P., Heinosaari, T.: Approximate joint measurements of qubit observables. Quantum Inf. Comput. 8, 0797–0818 (2008)

    MathSciNet  Google Scholar 

  4. Busch, P., Lahti, P.J., Mittelstaedt, P.: The Quantum Theory of Measurement, 2nd edn. Springer, Berlin (1996)

    MATH  Google Scholar 

  5. Heinosaari, T., Reitzner, D., Stano, P., Ziman, M.: Coexistence of quantum operations. J. Phys. A 42, 365302 (2009)

    Article  MathSciNet  Google Scholar 

  6. Heinosaari, T., Wolf, M.M.: Nondisturbing quantum measurements. J. Math. Phys. 51, 092201 (2010)

    Article  ADS  MathSciNet  Google Scholar 

  7. Heinosaari, T., Ziman, M.: The Mathematical Language of Quantum Theory. Cambridge University Press, Cambridge (2012)

    MATH  Google Scholar 

  8. Kraus, K.: States, Effects, and Operations. Springer, Berlin (1983)

    MATH  Google Scholar 

  9. Lahti, P.: Coexistence and joint measurability in quantum mechanics. Int. J. Theor. Phys. 42, 893–906 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  10. Lahti, P., Pulmannová, S.: Coexistent observables and effects in quantum mechanics. Rep. Math. Phys. 39, 339–351 (1997)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  11. Ozawa, M.: Quantum measuring processes of continuous observables. J. Math. Phys. 25, 79–87 (1984)

    Article  ADS  MathSciNet  Google Scholar 

  12. Ozawa, M.: Operations, disturbance, and simultaneous measurability. Phys. Rev. A 63, 032109 (2001)

    Article  ADS  Google Scholar 

  13. Paulsen, V.: Completely Bounded Maps and Operator Algebras. Cambridge University Press, Cambridge (2003)

    Book  Google Scholar 

  14. Raginsky, M.: Radon-Nikodym derivatives of quantum operations. J. Math. Phys. 44, 5003–5020 (2003)

    Article  ADS  MATH  MathSciNet  Google Scholar 

Download references

Acknowledgements

T. Heinosaari acknowledges financial support from the Academy of Finland (grant no. 138135). T. Miyadera acknowledges JSPS KAKENHI (grant no. 22740078). D. Reitzner acknowledges financial support from the project COQUIT.

Author information

Authors and Affiliations

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

    Teiko Heinosaari

  2. Department of Nuclear Engineering, Kyoto University, 6158540, Kyoto, Japan

    Takayuki Miyadera

  3. Department of Mathematics, Technische Universität München, 85748, Garching, Germany

    Daniel Reitzner

  4. Research Center for Quantum Information, Slovak Academy of Sciences, Dúbravská cesta 9, 845 11, Bratislava, Slovakia

    Daniel Reitzner

Authors
  1. Teiko Heinosaari
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Takayuki Miyadera
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Daniel Reitzner
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Daniel Reitzner.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Heinosaari, T., Miyadera, T. & Reitzner, D. Strongly Incompatible Quantum Devices. Found Phys 44, 34–57 (2014). https://doi.org/10.1007/s10701-013-9761-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10701-013-9761-1

Keywords

search

Navigation