A Markov chain is a tripartite quantum state ρ ABC where there exists a recovery map R B→BC such that ρ ABC = R B→BC (ρ AB ). More generally, an approximate Markov chain ρ ABC is a state whose distance to the closest recovered state R B→BC (ρ AB ) is small. Recently it has been shown that this distance can be bounded from above by the conditional mutual information I(A : C|B) ρ of the state. We improve on this connection by deriving the first bound that is tight in the commutative case and features an explicit recovery map that only depends on the reduced state pBC. The key tool in our proof is a multivariate extension of the Golden-Thompson inequality, which allows us to extend logarithmic trace inequalities from two to arbitrarily many matrices.