Skip to main content
SpringerLink
Log in

Gaussian dynamics equation in normal product form

  • Published:
Quantum Information Processing Aims and scope Submit manuscript

Abstract

In this paper, we discuss the normal product form of the density operator of multimode Gaussian states and obtain the correlation equation between the kernel matrix \(\textbf{R}\) of the Gaussian density operator in the norm al product form and its kernel matrix \(\textbf{G}\) in the standard quadratic form. Further, we explore the time evolution mechanism of \(\textbf{R}\) and obtain the Gaussian dynamical equation under the normal product \(\overset{ \cdot }{\textbf{R}}=i(\mathbf {RJH-HJR})\). Our work is devoted to searching for another mechanism for Gaussian dynamics. By exploring the description of the normal ordered density matrix under the coherent state representation, we find that our mechanism is feasible and easy to operate.

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

The data that support the findings of this study are available from the corresponding author upon reasonable request.

References

  1. Weedbrook, C., Pirandola, S., García-Patrón, R., Cerf, N.J., Ralph, T.C., Shapiro, J.H., Lloyd, S.: Gaussian quantum information. Rev. Mod. Phys. 84, 621 (2012)

    Article  ADS  Google Scholar 

  2. DiVincenzo, D.P., Terhal, B.M.: Fermionic linear optics revisited. Found. Phys. 35, 1967 (2005)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  3. Eisler, V., Zimborás, Z.: Entanglement negativity in two-dimensional free lattice models. N. J. Phys. 17, 053048 (2015)

    Article  Google Scholar 

  4. Greplová, E., Giedke, G.: Degradability of fermionic gaussian channels. Phys. Rev. Lett. 121, 200501 (2018)

    Article  ADS  Google Scholar 

  5. Kraus, C.V., Wolf, M.M., Cirac, J.I., Giedke, G.: Pairing in fermionic systems: a quantum-information perspective. Phys. Rev. A 79, 012306 (2009)

    Article  ADS  Google Scholar 

  6. A. Ferraro, S. Olivares, M. G. A. Paris, (2005), arXiv:quant-ph/0503237

  7. Paris, M.G.A., Genoni, M.G., Shammah, N., Teklu, B.: Quantifying the nonlinearity of a quantum oscillator. Phys. Rev. A 90, 012104 (2014)

    Article  ADS  Google Scholar 

  8. Onuma-Kalu, M., Grimmer, D.J., Mann, R.B., Martin-Martinez, E.: A classification of Markovian fermionic Gaussian master equations. J. Phys. A Math. Theor. 52, 435302 (2019)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  9. Grimmer, D., Brown, E., Kempf, A., Mann, R.B., Martin-Martinez, E.: A classification of open Gaussian dynamics. J. Phys. A Math. Theor. 51, 245301 (2018)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  10. A. Serafini, Quantum Continuous Variables: A Primer of Theoretical Methods, (CRC Press, 2017)

  11. Williamson, J.: On the algebraic problem concerning the normal forms of linear dynamical systems. Am. J. Math. 58, 141 (1936)

    Article  MathSciNet  MATH  Google Scholar 

  12. See for example, M. M. Wilde, Lecture notes on "Gaussian Quantum Information", (2015)

  13. He, R.: New algorithm to calculate the covariance matrix of an arbitrary form of Gaussian state. Quantum Inf. Process. 14, 3971 (2015)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  14. Parthasarathy, K.R.: What is a Gaussian state? Commun. Stoch. Anal. 4, 143 (2010)

    MathSciNet  MATH  Google Scholar 

  15. Wang, X.B., Hiroshima, T., Tomita, A., Hayashi, M.: Quantum information with Gaussian states. Phys. Rep. 448, 1 (2007)

    Article  ADS  MathSciNet  Google Scholar 

  16. Fan, H.Y.: Recent development of Dirac’s representation theory. In: Feng, D.H., Klauder, J.R., Strayer, M.R. (eds.) (Coherent states. Academic Press, New York (1994)

  17. Teretenkov, A.E.: Quadratic dissipative evolution of Gaussian states with drift. Math. Notes 101, 341 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  18. Lindblad, G.: On the generators of quantum dynamical semigroups. Commun. Math. Phys. 48, 119 (1976)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  19. Fan, H.Y., Klauder, J.R.: Canonical coherent-state representation of some squeeze operators. J Phys. A: Math. and Theor. 21, L725 (1988)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The work is supported by the School-level teaching and research project of West Anhui University (Grant wxxy2020047), Provincial Teaching and Research Projects of Higher Education Institutions in Anhui Province (Grant 2021jyxm1666) and The Natural Science Research Key Project of Education Department of Anhui Province of China (Grant No. KJ2021A0943).

Author information

Authors and Affiliations

  1. School of Electrical and Optoelectronic Engineering, West Anhui University, Lu’an, 237012, Anhui, China

    Rui He

Authors
  1. Rui He
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Rui He.

Ethics declarations

Conflict of interest

The author does not have any possible conflicts of interest.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

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

He, R. Gaussian dynamics equation in normal product form. Quantum Inf Process 22, 253 (2023). https://doi.org/10.1007/s11128-023-04015-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s11128-023-04015-z

Keywords

Search

Navigation