Communications in Mathematical Physics

, Volume 246, Issue 2, pp 359–374

Structure of States Which Satisfy Strong Subadditivity of Quantum Entropy with Equality


  • Patrick Hayden
    • Institute for Quantum Information
  • Richard Jozsa
    • Department of Computer ScienceUniversity of Bristol
  • Dénes Petz
    • Department for Mathematical AnalysisMathematical Institute, Budapest University of Technology and Economics
    • Department of Computer ScienceUniversity of Bristol
    • Department of MathematicsUniversity of Bristol, University Walk

DOI: 10.1007/s00220-004-1049-z

Cite this article as:
Hayden, P., Jozsa, R., Petz, D. et al. Commun. Math. Phys. (2004) 246: 359. doi:10.1007/s00220-004-1049-z


We give an explicit characterisation of the quantum states which saturate the strong subadditivity inequality for the von Neumann entropy. By combining a result of Petz characterising the equality case for the monotonicity of relative entropy with a recent theorem by Koashi and Imoto, we show that such states will have the form of a so–called short quantum Markov chain, which in turn implies that two of the systems are independent conditioned on the third, in a physically meaningful sense. This characterisation simultaneously generalises known necessary and sufficient entropic conditions for quantum error correction as well as the conditions for the achievability of the Holevo bound on accessible information.

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© Springer-Verlag Berlin Heidelberg 2004