Matrix product states and the decay of quantum conditional mutual information

A uniform matrix product state defined on a tripartite system of spins, denoted by $ABC,$ is shown to be an approximate quantum Markov chain when the size of subsystem $B,$ denoted $|B|,$ is large enough. The quantum conditional mutual information (QCMI) is investigated and proved to be bounded by a function proportional to $\exp(-q(|B|-K)+2K\ln|B|)$, with $q$ and $K$ computable constants. The properties of the bounding function are derived by a new approach, with a corresponding improved value given for its asymptotic decay rate $q$. We show the improved value of $q$ to be optimal. Numerical investigations of the decay of QCMI are reported for a collection of matrix product states generated by selecting the defining isometry with respect to Haar measure.