Skip to main content
SpringerLink
Log in

On positive operator-valued measures generated by a family of one-dimensional projectors

  • Original Paper
  • Published:
Annals of Functional Analysis Aims and scope Submit manuscript

Abstract

We study positive operator-valued measures generated by projections on one-dimensional subspaces. A special attention is paid to the case in which subspaces are spanned by vectors forming a Riesz basis. It is shown that the measurement fulfilled by such measure is informationally complete for quantum states being a convex hull of projections on subspaces spanned by the system of biorthogonal vectors. We also discuss the properties of different quantum channels associated with a discrete measurement. Finally, we show that our measurement allows to introduce a quantum instrument taking values in the set of two points.

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

Data availability statement

Data sharing is not applicable to this article as no new data were created or analyzed in this study.

References

  1. Alekseev, A.O., Amosov, G.G.: On restoring a quantum state after the measurement. Lobachevskii J. Math. 44(6), 1974–1979 (2023)

    Article  MathSciNet  Google Scholar 

  2. Alekseev, A.O., Amosov, G.G.: On extension of the family of projections to positive operator-valued measure. Vestn. St. Petersbg. Univ. Math. 56(1), 1–8 (2023)

    Article  MathSciNet  Google Scholar 

  3. Amosov, G.G., Mancini, S., Man’ko, V.I.: On the information completeness of quantum tomograms. Phys. Lett. A 372(16), 2820–2824 (2008)

    Article  MathSciNet  Google Scholar 

  4. Appleby, D.M.: Symmetric informationally complete-positive operator valued measures and the extended Clifford group. J. Math. Phys. 46, 052107 (2005)

    Article  MathSciNet  Google Scholar 

  5. Busch, P.: Informationally complete sets of physical quantities. Int. J. Theor. Phys. 30, 1217 (1991)

    Article  MathSciNet  Google Scholar 

  6. Casazza, P.G.: The art of frame theory. Taiwan. J. Math. 4(2), 129–201 (2000)

    Article  MathSciNet  Google Scholar 

  7. D’Ariano, G.M., Perinotti, P., Sacchi, M.F.: Informationally complete measurements and group representation. J. Opt. B Quant. Semiclass. Opt. 6(6), S487 (2004)

    Article  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  9. Durt, T., Englert, B.-G., Bengtsson, I., Zyczkowski, K.: On mutually unbiased bases. Int. J. Quant. Inf. 8, 535–640 (2010)

    Article  Google Scholar 

  10. Helstrom, C.W.: Quantum detection and discrimination theory. J. Stat. Phys. 1, 231–252 (1969)

    Article  Google Scholar 

  11. Holevo, A.S.: On complementary channels and the additivity problem. Probab. Theory Appl. 51, 133–143 (2005)

    MathSciNet  Google Scholar 

  12. Holevo, A.: Probabilistic and Statistical Aspects of Quantum Theory. Publications of the Scuola Normale Superiore, Pisa (2011)

    Book  Google Scholar 

  13. Holevo, A.S.: Quantum System, Channels, Information. De Gruyter, Berlin (2013)

    Google Scholar 

  14. Holevo, A.: On the classical capacity of general quantum Gaussian measurement. Entropy 23(3), 377 (2021)

    Article  MathSciNet  Google Scholar 

  15. Ivanovic, I.D.: How to differentiate between non-orthogonal states. Phys. Lett. A 126(6), 257–259 (1987)

    Article  Google Scholar 

  16. Kronberg, D.A.: Increasing the distinguishability of quantum states with an arbitrary success probability. Proc. Steklov Inst. Math. 313, 113–119 (2021)

    Article  MathSciNet  Google Scholar 

  17. Kronberg, D.A.: Modification of quantum measurements by mapping onto quantum states and classical outcomes. Lobachevskii J. Math. 43(7), 1663–1668 (2022)

    Article  MathSciNet  Google Scholar 

  18. Marcus, A.W., Spielman, D.A., Srivastava, N.: Interlacing families II: mixed characteristic polynomials and the Kadison–Singer problem. Ann. Math. 182(1), 327–350 (2015)

    Article  MathSciNet  Google Scholar 

  19. Peres, A.: How to differentiate between non-orthogonal states. Phys. Lett. A 128(1–2), 19 (1988)

    Article  MathSciNet  Google Scholar 

  20. Prugovecki, E.: Information-theoretical aspects of quantum measurement. Int. J. Theor. Phys. 16, 321 (1977)

    Article  Google Scholar 

  21. Renes, J.M., Blume-Kohout, R., Scott, A.J., Caves, C.M.: Symmetric informationally complete quantum measurements. J. Math. Phys. 45(6), 2171–2180 (2004)

    Article  MathSciNet  Google Scholar 

  22. Schroek, F.E.: Quantum Mechanics on Phase Space. Kluwer, Dordrecht (1996)

    Book  Google Scholar 

Download references

Acknowledgements

The work of G.G. Amosov was performed at the Steklov International Mathematical Center and supported by the Ministry of Science and Higher Education of the Russian Federation (agreement no. 075-15-2022-265).

Author information

Authors and Affiliations

  1. Steklov Mathematical Institute of Russian Academy of Sciences, Moscow, Russia

    G. G. Amosov & D. A. Kronberg

  2. St. Petersburg State University, St. Petersburg, Russia

    A. D. Baranov

Authors
  1. G. G. Amosov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. A. D. Baranov
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. D. A. Kronberg
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to G. G. Amosov.

Additional information

Communicated by Airat Bikchentaev.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Amosov, G.G., Baranov, A.D. & Kronberg, D.A. On positive operator-valued measures generated by a family of one-dimensional projectors. Ann. Funct. Anal. 15, 48 (2024). https://doi.org/10.1007/s43034-024-00351-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s43034-024-00351-y

Keywords

Mathematics Subject Classification

Search

Navigation