The free contraction norm was introduced by Belinschi, Collins and Nechita as a tool to compute the typical location of the collection of singular values associated to a random subspace of the tensor product of two Hilbert spaces. In turn, it was used by them to obtain sharp bounds for the violation of the additivity of the minimum output entropy for random quantum channels with Bell states. In this talk, I will discuss some free probability techniques involved in the study of free contraction norm and their applications to the additivity violations. The talk is based on joint work with B. Collins and M. Fukuda.