Communications in Mathematical Physics

, Volume 259, Issue 1, pp 129–138 | Cite as

A New Inequality for the von Neumann Entropy

Article

Abstract

Strong subadditivity of von Neumann entropy, proved in 1973 by Lieb and Ruskai, is a cornerstone of quantum coding theory. All other known inequalities for entropies of quantum systems may be derived from it. Here we prove a new inequality for the von Neumann entropy which we prove is independent of strong subadditivity: it is an inequality which is true for any four party quantum state, provided that it satisfies three linear relations (constraints) on the entropies of certain reduced states.

Keywords

Entropy Neural Network Statistical Physic Complex System Nonlinear Dynamics 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Accardi, L., Frigerio, A.: Markovian cocycles. Proc. Roy. Irish Acad. 83A(2), 251–263 (1983)Google Scholar
  2. 2.
    Araki, H., Lieb, E.H.: Entropy inequalities. Commun. Math. Phys. 18, 160–170 (1970)CrossRefGoogle Scholar
  3. 3.
    Bennett, C.H., Shor, P.W.: Quantum information theory. IEEE Trans. Inf. Theory 44(6), 2724–2742 (1998)CrossRefGoogle Scholar
  4. 4.
    Hayden, P., Jozsa, R., Petz, D., Winter, A.: Structure of states which satisfy strong subadditivity of quantum entropy with equality. Commun. Math. Phys. 246(2), 359–374 (2004)CrossRefGoogle Scholar
  5. 5.
    Lieb, E.H.: Some Convexity and Subadditivity Properties of Entropy. Bull. Amer. Math. Soc. 81, 1–13 (1975)Google Scholar
  6. 6.
    Lieb, E.H., Ruskai, M.B.: Proof of the strong subadditivity of quantum-mechanical entropy. J. Math. Phys. 14, 1938–1941 (1973)CrossRefGoogle Scholar
  7. 7.
    Linden, N., Maneva, E., Massar, S., Popescu, S., Roberts, D., Schumacher, B., Smolin, J.A., Thapliyal, A.V.: In preparationGoogle Scholar
  8. 8.
    Pippenger, N.: What are the laws of information theory? 1986 Special Problems in Communication and Computation Conference, Palo Alto, CA, 3–5 September 1986Google Scholar
  9. 9.
    Pippenger, N.: The inequalities of quantum information theory. IEEE Trans. Inf. Theory 49(4), 773–789 (2003)CrossRefGoogle Scholar
  10. 10.
    Ruskai, M.B.: Inequalities for quantum entropy: A review with conditions for equality. J. Math. Phys. 43(9), 4358–4375 (2002)CrossRefGoogle Scholar
  11. 11.
    Shannon, C.E.: A Mathematical Theory of Communication. Bell System Tech. J. 27, 379–423 and 623–656 (1948) Shannon Theory demi-centennial issue of IEEE Trans. Inf. Theory: 44(6), (1998)Google Scholar
  12. 12.
    Yeung, R.W.: A Framework for Linear Information Inequalities. IEEE Trans. Inf. Theory 43(6), 11924–1934 (1997)Google Scholar
  13. 13.
    Zhang, Z., Yeung, R.W.: A Non-Shannon Type Conditional Inequality of Information Quantities. IEEE Trans. Inf. Theory 43(6), 1982–1985 (1997)CrossRefGoogle Scholar
  14. 14.
    Yeung, R.W., Zhang, Z.: On Characterization of Entropy Function via Information Inequalities. IEEE Trans. Inf. Theory 44(4), 1440–1452 (1998)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of Bristol, University WalkU.K

Personalised recommendations