Almost One Bit Violation for the Additivity of the Minimum Output Entropy

In a previous paper, we proved that, in the appropriate asymptotic regime, the limit of the collection of possible eigenvalues of output states of a random quantum channel is a deterministic, compact set K (k,t) . We also showed that the set K (k,t) is obtained, up to an intersection, as the unit ball of the dual of a free compression norm. In this paper, we identify the maximum of norms on the set K (k,t) and prove that the maximum is attained on a vector of shape (a, b, . . . , b) where a > b. In particular, we compute the precise limit value of the minimum output entropy of a single random quantum channel. As a corollary, we show that for any , it is possible to obtain a violation for the additivity of the minimum output entropy for an output dimension as low as 183, and that for appropriate choice of parameters, the violation can be as large as . Conversely, our result implies that, with probability one in the limit, one does not obtain a violation of additivity using conjugate random quantum channels and the Bell state, in dimension 182 and less.