  We make a number of simplifications in Gour and Friedland's proof of local
additivity of minimum output entropy of a quantum channel. We follow them in
reframing the question as one about entanglement entropy of bipartite states
associated with a $d_B \times d_E $ matrix. We use a different approach to
reduce the general case to that of a square positive definite matrix. We use
the integral representation of the log to obtain expressions for the first and
second derivatives of the entropy, and then exploit the modular operator and
functional calculus to streamline the proof following their underlying
strategy. We also extend this result to the maximum relative entropy with
respect to a fixed reference state which has important implications for
studying the superadditivity of the capacity of a quantum channel to transmit
classical information.
