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

TLDR: In this paper, the authors give an explicit characterisation of the quantum states which saturate the strong subadditivity inequality for the von Neumann entropy, and 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.

Abstract: 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.