Approximate unitary $n^{2/3}$-designs give rise to quantum channels with super additive classical Holevo capacity

In a breakthrough, Hastings' showed that there exist quantum channels whose classical capacity is superadditive i.e. more classical information can be transmitted by quantum encoding strategies entangled across multiple channel uses as compared to unentangled quantum encoding strategies. Hastings' proof used Haar random unitaries to exhibit superadditivity. In this paper we show that a unitary chosen uniformly at random from an approximate $n^{2/3}$-design gives rise to a quantum channel with superadditive classical Holevo capacity, where $n$ is the dimension of the unitary exhibiting the Stinespring dilation of the channel superoperator. We prove a sharp Dvoretzky-like theorem (similar to Aubrun, Szarek, Werner, 2010) stating that, with high probability under the choice of a unitary from an approximate $t$-design, random subspaces of large dimension make a Lipschitz function take almost constant value. Such theorems were known earlier only for Haar random unitaries. We obtain our result by appealing to Low's technique for proving concentration of measure for an approximate $t$-design, combined with a stratified analysis of the variational behaviour of Lipschitz functions on the unit sphere in high dimension. The stratified analysis is the main technical advance of this work. Finally we also show that for any $p>1$, approximate unitary $(n^{1.7} \log n)$-designs give rise to channels violating subadditivity of R\'{e}nyi $p$-entropy. In addition to stratified analysis, the proof of this result uses a new technique of approximating a monotonic differentiable function defined on a closed bounded interval and its derivative by moderate degree polynomials which should be of independent interest. Hence, our work can be viewed as a partial derandomisation of Hastings' result and a step towards the quest of finding an explicit quantum channel with superadditive classical Holevo capacity.