Quantum conditional mutual information and approximate Markov chains

Abstract

A state on a tripartite quantum system $A \otimes B \otimes C$ forms a Markov chain if it can be reconstructed from its marginal on $A \otimes B$ by a quantum operation from $B$ to $B \otimes C$. We show that the quantum conditional mutual information $I(A: C | B)$ of an arbitrary state is an upper bound on its distance to the closest reconstructed state. It thus quantifies how well the Markov chain property is approximated.

v3: 31 pages, corrected error in dimension factor for the application to squashed entanglement (Corollary D.3) and added some clarifications and remarks. v2: 30 pages, improved introduction and added application to squashed entanglement