Approximate Unitary $n^{2/3}$-Designs Give Rise to Quantum Channels with Superadditive Classical Holevo Capacity

Abstract: In a breakthrough, Hastings(2009) showed that there exists quantum channels whose classical Holevo 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 talk we will 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 do so by showing that the minimum output von Neumann entropy of a quantum channel arising from an approximate unitary $n^{2/3}$-design is subadditive, which implies superadditivity of classical Holevo capacity of quantum channels (Shor, 2004).

We follow the geometric functional analytic approach of Aubrun, Szarek and Werner(2010) in order to prove our result. More precisely we prove a sharp Dvoretzky-like theorem 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 (2009}) 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. Haar random unitaries require at least $\Omega(n^2)$ random bits in order to describe them with good precision. In contrast, there exist exact $n^{2/3}$-designs using only $O(n^{2/3} \log n)$ random bits (Kuperberg, 2006). Thus, 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.