Quantum marginal problems

Topical Review

Abstract

The quantum marginal problem asks the question if elements of a given set of quantum states can be reduced states of some joint quantum state. In this paper we present various versions of the quantum marginal problem along with their solutions in the order in which they were published. The review begins with simple finite-dimensional composite systems and ends with results of the Gaussian quantum marginal problem that apply to systems of harmonic oscillators (e.g. modes of the quantum electromagnetic field).

Graphical abstract

Keywords

Quantum Information 

References

  1. 1.
    S. Bravyi, Quant. Inf. Comput. 4, 12 (2004)MATHMathSciNetGoogle Scholar
  2. 2.
    A. Higuchi, A. Sudbery, J. Szulc, Phys. Rev. Lett. 90, 107902 (2003)MathSciNetCrossRefADSGoogle Scholar
  3. 3.
    A. Higuchi, e-print arXiv:quant-ph/0309186 (2003)Google Scholar
  4. 4.
    A.A. Klyachko, e-print arXiv:quant-ph/0409113 (2004)Google Scholar
  5. 5.
    J. Eisert, T. Tyc, T. Rudolph, B.C. Sanders, Commun. Math. Phys. 280, 263 (2008)MATHMathSciNetCrossRefADSGoogle Scholar
  6. 6.
    C. Schilling, e-print arXiv:quant-ph/1404.1085 (2014)Google Scholar
  7. 7.
    A.A. Klyachko, J. Phys.: Conf. Ser. 36, 72 (2006)ADSGoogle Scholar
  8. 8.
    M. Altunbulak, A.A. Klyachko, Commun. Math. Phys. 282, 287 (2008)MATHMathSciNetCrossRefADSGoogle Scholar
  9. 9.
    C. Schilling, D. Gross, M. Christandl, Phys. Rev. Lett. 110, 040404 (2013)CrossRefADSGoogle Scholar
  10. 10.
    Y.-K. Liu, M. Christandl, F. Verstraete, Phys. Rev. Lett. 98, 110503 (2007)CrossRefADSGoogle Scholar
  11. 11.
    E.A. Carlen, J.L. Lebowitz, E.H. Lieb, J. Math. Phys. 54, 062103 (2013)MathSciNetCrossRefADSGoogle Scholar
  12. 12.
    J. Chen, Z. Ji, D. Kribs, N. Lütkenhaus, B. Zeng, Phys. Rev. A 90, 032318 (2014)CrossRefADSGoogle Scholar
  13. 13.
    T.J. Osborne, e-print arXiv:quant-ph/0806.2962v1 (2008)Google Scholar
  14. 14.
    L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, 1995)Google Scholar
  15. 15.
    R. Simon, N. Mukunda, Biswadeb Dutta, Phys. Rev. A 49, 1567 (1994)MathSciNetCrossRefADSGoogle Scholar
  16. 16.
    G. Vidal, R.F. Werner, Phys. Rev. A 65, 032314 (2002)CrossRefADSGoogle Scholar
  17. 17.
    A. Botero, B. Reznik, Phys. Rev. A 67, 052311 (2003)CrossRefADSGoogle Scholar
  18. 18.
    G. Adesso, A. Serafini, F. Illuminati, Phys. Rev. A 73, 032345 (2006)CrossRefADSGoogle Scholar
  19. 19.
    M. Krbek, T. Tyc, J. Vlach, J. Math. Phys. 55, 062201 (2014)MathSciNetCrossRefADSGoogle Scholar
  20. 20.
    S.L. Braunstein, Phys. Rev. A 71, 055801 (2005)CrossRefADSGoogle Scholar

Copyright information

© EDP Sciences, SIF, Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.Department of Theoretical Physics and Astrophysics, Faculty of ScienceMasaryk UniversityBrnoCzech Republic
  2. 2.Department of Computer Science, Faculty of InformaticsMasaryk UniversityBrnoCzech Republic

Personalised recommendations