Skip to main content
SpringerLink
Log in

Symplectic dilations, Gaussian states and Gaussian channels

  • Published:
Indian Journal of Pure and Applied Mathematics Aims and scope Submit manuscript

Abstract

By elementary matrix algebra we show that every real 2n×2n matrix admits a dilation to an element of the real symplectic group Sp(2(n+m)) for some nonnegative integer m. Our methods do not yield the minimum value of m, for which such a dilation is possible.

After listing some of the main properties of Gaussian states in L 2(Rn), we analyse the implications of symplectic dilations in the study of quantum Gaussian channels which lead to some interesting open problems, particularly, in the context of the work of Heinosaari et al., [3].

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

References

  1. B. Arvind Dutta, N. Mukunda and R. Simon, The real symplectic group in quantum mechanics and optics, Pramana-J. Phys., 45 (1995), 471–497.

    Article  Google Scholar 

  2. F. Caruso, J. Eisert, V. Giovannetti and A. S. Holevo, Multimode bosonic Gaussian channels, New J. Phys., 10:083030 (2008).

    Article  Google Scholar 

  3. T. Heinosaari, A. S. Holevo and M. M. Wolf, The semigroup structure of Gaussian channels, arXiv: 0909.0408v1 [quant-ph] 2 Sep 2009, J. Quantum Inf. Comp., 10:0619–0635 (2010).

    MathSciNet  Google Scholar 

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

    MATH  Google Scholar 

  5. A. S. Holevo and R. F. Werner, Evaluating capacities of bosonic Gaussian channels, Phys. Rev. A, 63:032312 (2001).

    Article  Google Scholar 

  6. K. R. Parthasarathy, What is a Gaussian state? Commun. Stoch. Anal., 4 (2010), 143–160.

    MathSciNet  Google Scholar 

  7. K. R. Parthasarathy, The symmetry group of Gaussian states in L 2(Rn), in iProkhorov and Contemporary Probability 349-369 (2013).Eds: Shiryaev A. N., Varadhan S. R. S. and Presman E. L. (Berlin, Springer Proceedings in Mathematics and Statistics 33).

  8. A. Soifer, Mathematics of Colouring and the colourful Life of its Creators (2009).(New York: Springer Verlag).

    Google Scholar 

  9. P. Vanheuverzwijn, Generators for completely positive semigroups, Ann. Inst. H. Poincaré Sect. A (N.S.), 29 (1978), 123–138.

    MATH  MathSciNet  Google Scholar 

  10. J. Williamson, The exponential representation of canonical matrices, Amer. J. Math., 61 (1939), 897–911.

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Indian Statistical Institute, Delhi Centre, 7, S. J. S. Sansanwal Marg, New Delhi, 110 016, India

    K. R. Parthasarathy

Authors
  1. K. R. Parthasarathy
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to K. R. Parthasarathy.

Additional information

Dedicated to Professor Kalyan B. Sinha on his 70th birthday.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Parthasarathy, K.R. Symplectic dilations, Gaussian states and Gaussian channels. Indian J Pure Appl Math 46, 419–439 (2015). https://doi.org/10.1007/s13226-015-0144-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s13226-015-0144-5

Keywords

Search

Navigation