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Covariant CP-Instruments and Their Convolution Semigroups

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Abstract

Using probability operators and Fourier transforms of CP-instruments on von Neumann algebras, we give necessary and sufficient conditions for operators to be probability operators associated with covariant CP-instruments or to be Fourier transforms of covariant CP-instruments. We discuss a convolution semigroup of covariant CP-instruments and a semigroup of probability operators associated with CP-instruments on von Neumann algebras.

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References

  1. Barchielli, A.: Probability operators and convolution semigroups of instruments in quantum probability. Probab. Theory Relat. Fields 82, 1–8 (1989)

    Article  MathSciNet  Google Scholar 

  2. Barchielli, A.: Some Markov semigroups in quantum probability. In: Accardi, L., von Waldenfels, W. (eds.) Quantum Probability and Applications, V (Heidelberg, 1988). Lecture Notes in Mathematics, vol. 1442, pp. 86–98. Springer, Berlin (1990)

  3. Barchielli, A., Holevo, A.S., Lupieri, G.: An analogue of Hunt’s representation theorem in quantum probability. J. Theor. Probab. 6, 231–265 (1993)

  4. Carmeli, C., Heinosaari, T., Toigo, A.: Covariant quantum instruments. J. Funct. Anal. 257, 3353–3374 (2009)

    Article  MathSciNet  Google Scholar 

  5. Davies, E.: On the repeated measurements of continuous observables in quantum mechanics. J. Funct. Anal. 6, 318–346 (1970)

    Article  MathSciNet  Google Scholar 

  6. Davies, E.: Quantum Theory of Open Systems. Academic Press, New York (1976)

    MATH  Google Scholar 

  7. Davies, E., Lewis, J.: An operational approach to quantum probability. Commun. Math. Phys. 17, 239–260 (1970)

    Article  MathSciNet  Google Scholar 

  8. Denisov, L.V.: On the Stinespring-type theorem for covariant instruments in noncommutative probability theory. Reports on mathematics and its applications of the Steklov Mathematical Institute, vol. 20, Moscow (1990)

  9. Haapasalo, E., Pellonpää, J.-P.: Covariant KSGNS construction and quantum instruments. Rev. Math. Phys. 29, 1750020, 47 (2017)

    Article  MathSciNet  Google Scholar 

  10. Haapasalo, E., Pellonpää, J.-P.: Optimal covariant quantum measurements. J. Phys. A: Math. Theor. 54, 155304, 50 (2021)

    Article  MathSciNet  Google Scholar 

  11. Heo, J., Belavkin, V., Ji, U.C.: Monotone quantum stochastic processes and covariant dynamical hemigroups. J. Funct. Anal. 261, 3345–3365 (2011)

    Article  MathSciNet  Google Scholar 

  12. Heo, J., Belavkin, V., Ji, U.C.: Reconstruction theorem for stationary monotone quantum Markov processes. Bull. Korean Math. Soc. 49, 63–74 (2012)

    Article  MathSciNet  Google Scholar 

  13. Heyer, H.: Probability Measures on Locally Compact Groups. Springer, Berlin (1977)

    Book  Google Scholar 

  14. Holevo, A.S.: Radon–Nikodym derivatives of quantum instruments. J. Math. Phys. 39, 1373–1387 (1998)

  15. Holevo, A.S.: Probabilistic and Statistical Aspects of Quantum Theory, 2nd edn. Quaderni/Monographs, 1. Edizioni della Normale, Pisa (2011)

  16. Jauch, J., Piron, C.: Generalized localizability. Helv. Phys. Acta 40, 559–570 (1967)

    MATH  Google Scholar 

  17. Lau, A.: The Fourier–Stieltjes algebra of a topological semigroup with involution. Pac. J. Math. 77, 165–181 (1978)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  19. Ylinen, K.: Quantum instruments and related transformation valued functions. Found. Phys. 39, 656–675 (2009)

    Article  MathSciNet  Google Scholar 

Download references

Funding

This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by MSIT.

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Authors and Affiliations

  1. Department of Mathematics, Research Institute for Natural Sciences, Hanyang University, Seoul, 04763, Korea

    Jaeseong Heo

  2. Department of Mathematics, Institute for Industrial and Applied Mathematics, Chungbuk National University, Cheongju, 28644, Korea

    Un Cig Ji

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  1. Jaeseong Heo
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  2. Un Cig Ji
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Correspondence to Jaeseong Heo.

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Communicated by Jussi Behrndt, Fabrizio Colombo and Sergey Naboko.

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This article is part of the topical collection “Spectral Theory and Operators in Mathematical Physics” edited by Jussi Behrndt, Fabrizio Colombo and Sergey Naboko.

Jaeseong Heo: This was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (No. 2020R1A4A3079066). Un Cig Ji: This work was supported by Basic Science Research Program through the NRF funded by the MEST (NRF-2016R1D1A1B01008782). Spectral Theory and Operators in Mathematical Physics.

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Heo, J., Ji, U.C. Covariant CP-Instruments and Their Convolution Semigroups. Complex Anal. Oper. Theory 15, 98 (2021). https://doi.org/10.1007/s11785-021-01143-1

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