  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.
