Using the graphical calculus and integration techniques introduced by the
authors, we study the statistical properties of outputs of products of random
quantum channels for entangled inputs. In particular, we revisit and generalize
models of relevance for the recent counterexamples to the minimum output
entropy additivity problems. Our main result is a classification of regimes for
which the von Neumann entropy is lower on average than the elementary bounds
that can be obtained with linear algebra techniques
