It is widely known in the
quantum information community that the states that satisfy strong subadditivity
of entropy with equality have the form of quantum Markov chain. Based on a
recent strengthening of strong subadditivity of entropy, I will describe how
such structure can be exploited in the studies of gapped quantum many-body
system. In particular, I will describe a diagrammatic trick to i) give a
quantitative statement about the locality of entanglement spectrum ii)
perturbatively bound changes of topological entanglement entropy under generic
perturbation.
