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Journal of Russian Laser Research

, Volume 35, Issue 5, pp 457–461 | Cite as

New Inequality for Density Matrices of Single Qudit States

  • Vladimir N. Chernega
  • Olga V. Man’ko
  • Vladimir I. Man’ko
Article

Abstract

Using the monotonicity of relative entropy of composite quantum systems, we obtain new entropic inequalities for arbitrary density matrices of single qudit states. Examples of qutrit state inequalities and the “qubit portrait” bound for the distance between the qutrit states are considered in explicit form.

Keywords

Hermitian matrix composite quantum system noncomposite quantum system entropic and information inequalities 

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References

  1. 1.
    M. A. Nielsen, and I. L. Chuang, Quantum Computation and Quantum Information, Cambridge University Press, Cambridge, UK (2000).MATHGoogle Scholar
  2. 2.
    E. H. Lieb and M. B. Ruskai, J. Math. Phys., 14, 1938 (1973).ADSCrossRefMathSciNetGoogle Scholar
  3. 3.
    M. A. Nielsen and D. Petz, “A simple proof of the strong subadditivity inequality,” arXiv:quantph/ 0408130 (2005).Google Scholar
  4. 4.
    E. A. Carlen, and E. H. Lieb, Lett. Math. Phys., 83, 107 (2008) [arXiv:0710.4167 (2007)].ADSCrossRefMATHMathSciNetGoogle Scholar
  5. 5.
    E. A. Carlen and Elliott H. Lieb, “A Minkowski-type trace inequality and strong subadditivity of quantum entropy,” arXiv:math/0701352 (2007).Google Scholar
  6. 6.
    H. Araki and E. H. Lieb, Commun. Math. Phys., 18, 160 (1970).ADSCrossRefMathSciNetGoogle Scholar
  7. 7.
    E. A. Carlen and E. H. Lieb, Lett. Math. Phys., 101, 1 (2012).ADSCrossRefMATHMathSciNetGoogle Scholar
  8. 8.
    E. H. Lieb, Adv. Math., 11, 267 (1973).CrossRefMATHMathSciNetGoogle Scholar
  9. 9.
    E. H. Lieb, Bull. Am. Math. Soc., 81, 1 (1975).CrossRefMATHMathSciNetGoogle Scholar
  10. 10.
    M. B. Ruskai, J. Math. Phys., 43, 4358 (2002); erratum ibid 46, 019901 (2005).ADSCrossRefMATHMathSciNetGoogle Scholar
  11. 11.
    M. B Ruskai, Commun. Math. Phys., 26, 280 (1972).Google Scholar
  12. 12.
    E. A. Carlen and E. H. Lieb, “Remainder terms for some quantum entropy inequalities,” arXiv:1402.3840 [quant-ph] (2014).Google Scholar
  13. 13.
    D. W. Robinson and D. Ruelle, Commun. Math. Phys., 5, 288 (1967).ADSCrossRefMATHMathSciNetGoogle Scholar
  14. 14.
    O. Lanford III and D. W. Robinson, J. Math. Phys., 9, 1120 (1968).ADSCrossRefMATHMathSciNetGoogle Scholar
  15. 15.
    M. B. Ruskai, “Lieb’s simple proof of concavity (A,B) TrA p K B 1−p K and remarks on related inequalities,” arXiv:quant-ph/0404126v1 (2004).Google Scholar
  16. 16.
    V. N. Chernega and O. V. Man’ko, J. Russ. Laser Res., 34, 383 (2013).CrossRefGoogle Scholar
  17. 17.
    V. N. Chernega and O. V. Man’ko, J. Russ. Laser Res., 35, 27 (2014).CrossRefGoogle Scholar
  18. 18.
    M. A. Man’ko and V. I. Man’ko, Phys. Scr., T160, 014030 (2014).ADSCrossRefGoogle Scholar
  19. 19.
    M. A. Man’ko and V. I. Man’ko, “The maps of matrices and portrait maps of density operators of composite and noncomposite systems,” arXiv:1404.3650 [quant-ph] (2014).Google Scholar
  20. 20.
    V. N. Chernega, O. V. Man’ko, and V. I. Man’ko, “Minkowski-type inequality for arbitrary density matrix of composite and noncomposite systems,” arXiv:1405-4956v1 [quant-ph] (2014).Google Scholar
  21. 21.
    V. N. Chernega and V. I. Man’ko, J. Russ. Laser Res., 28, 103 (2007).CrossRefGoogle Scholar
  22. 22.
    C. Lupo, V. I. Man’ko, and G. Marmo, J. Phys. A: Math. Theor., 40, 13091 (2007).ADSCrossRefMATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  • Vladimir N. Chernega
    • 1
  • Olga V. Man’ko
    • 1
  • Vladimir I. Man’ko
    • 1
    • 2
  1. 1.P. N. Lebedev Physical InstituteRussian Academy of SciencesMoscowRussia
  2. 2.Moscow Institute of Physics and Technology (State University)DolgoprudnyíRussia

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