  The problem of additivity of the Minimum Output Entropy is of fundamental
importance in Quantum Information Theory (QIT). It was solved by Hastings in
the one-shot case, by exhibiting a pair of random quantum channels. However,
the initial motivation was arguably to understand regularized quantities and
there was so far no way to solve additivity questions in the regularized case.
The purpose of this paper is to give a solution to this problem. Specifically,
we exhibit a pair of quantum channels which unearths additivity violation of
the regularized minimum output entropy. Unlike previously known results in the
one-shot case, our construction is non-random, infinite dimensional and in the
commuting-operator setup. The commuting-operator setup is equivalent to the
tensor-product setup in the finite dimensional case for this problem, but their
difference in infinite dimensional setting has attracted substantial attention
and legitimacy recently in QIT with the celebrated resolutions of Tsirelson's
and Connes embedding problem. Likewise, it is not clear that our approach works
in the finite dimensional setup. Our strategy of proof relies on developing a
variant of the Haagerup inequality optimized for a product of free groups.
