Citation: Hayden, Patrick, et al. "Structure of states which satisfy strong subadditivity of quantum entropy with equality." Communications in mathematical physics 246 (2004): 359-374.
Web: https://arxiv.org/abs/quant-ph/0304007
Tags: Information-theory, Computer-scientific
This is the first paper to study quantum Markov states. The authors take a good look at tripartite states which saturate the strong subadditivity of quantum entropy. They show that these states have vanishing conditional mutual information, they satisfy a Markov chain-type condition, and they can be expressed in a standard form via a structure theorem. All of these properties are put into context, and are explained in analogy with the classical case.
A key tool in the technical part of the work is a recent theorem of Petz:
> Petz, Dénes. "Sufficiency of channels over von Neumann algebras." The Quarterly Journal of Mathematics 39.1 (1988): 97-108.
> Petz, Dénes. "Monotonicity of quantum relative entropy revisited." Reviews in Mathematical Physics 15.01 (2003): 79-91.
This theorem gives a neccecary and sufficient condition for the monotonicity of quantum relative entropy to be saturated. These neccecary and sufficient conditions can then be translated (with the help of a theorem of Koashi and Imoto) to get the main structure theorem.
In words, the structure theorem says that quantum markov states are all classical probability distributions over states which split as a tensor product over an enlargement of A and an enlargement of C. This means that there is a measurement that can be performed which destroys no quantum information which breaks all the entanglement between A and C.