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Extreme Covariant Observables for Type I Symmetry Groups

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Abstract

The structure of covariant observables—normalized positive operator measures (POMs)—is studied in the case of a type I symmetry group. Such measures are completely determined by kernels which are measurable fields of positive semidefinite sesquilinear forms. We produce the minimal Kolmogorov decompositions for the kernels and determine those which correspond to the extreme covariant observables. Illustrative examples of the extremals in the case of the Abelian symmetry group are given.

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References

  1. Carmeli, C., Heinosaari, T., Pellonpää, J.-P., Toigo, A.: Extremal covariant positive operator valued measures: The case of a compact symmetry group. J. Math. Phys. 49, 063504 (2008)

    Article  ADS  MathSciNet  Google Scholar 

  2. Cattaneo, U.: On Mackey’s imprimitivity theorem. Comment. Math. Helv. 54, 629–641 (1979)

    Article  MATH  MathSciNet  Google Scholar 

  3. Chiribella, G., D’Ariano, G.M.: Extremal covariant positive operator valued measures. J. Math. Phys. 45, 4435–4447 (2004)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  4. D’Ariano, G.M.: Extremal covariant quantum operations and positive operator valued measures. J. Math. Phys. 45, 3620–3635 (2004)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  5. Dixmier, J.: C *-Algebras. North-Holland, Amsterdam (1977)

    MATH  Google Scholar 

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

    MATH  Google Scholar 

  7. Holevo, A.S.: Generalized imprimitivity systems for Abelian groups. Sov. Math. (Iz. VUZ) 27, 53–80 (1983)

    Google Scholar 

  8. Holevo, A.S.: Covariant measurements and imprimitivity systems. In: Lecture Notes in Mathematics, vol. 1055, pp. 153–172 (1984)

  9. Holevo, A.S.: On a generalization of canonical quantization. Math. USSR Izv. 28, 175–188 (1987)

    Article  Google Scholar 

  10. Kiukas, J., Pellonpää, J.-P.: A note on infinite extreme correlation matrices. Linear Algebra Appl. 428, 2501–2508 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  11. Li, C.-K., Tam, B.-S.: A note on extreme correlation matrices. SIAM J. Matrix Anal. Appl. 15, 903–908 (1994)

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Probability Theory, Steklov Mathematical Institute, Gubkina 8, 119991, Moscow, Russia

    Alexander S. Holevo

  2. Department of Physics, University of Turku, 20014, Turku, Finland

    Juha-Pekka Pellonpää

Authors
  1. Alexander S. Holevo
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  2. Juha-Pekka Pellonpää
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Correspondence to Juha-Pekka Pellonpää.

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Dedicated to Pekka J. Lahti in honor of his sixtieth birthday

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Holevo, A.S., Pellonpää, JP. Extreme Covariant Observables for Type I Symmetry Groups. Found Phys 39, 625–641 (2009). https://doi.org/10.1007/s10701-009-9274-0

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  • DOI: https://doi.org/10.1007/s10701-009-9274-0

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