The goal of this note is to show that Hastings' counterexample to the
additivity of minimal output von Neumann entropy can be
readily deduced from a sharp version of Dvoretzky's theorem.
The minimal output entropy is intimately related to the capacity of a quantum channel to transmit classical information. Hastings' example show that this capacity is non-additive, a phenomenon with no classical analogue. It is a challenging open problem to find a tractable formula for the classical capacity of a quantum channel.
Our approach requires a simple bootstraping argument in addition to the standard Dvoretzky theorem (with sharp dependence on epsilon) which might be of interest in other situations.
This is a continuation to the previous paper, where we use Dvoretzky's theorem to revisit Hayden-Winter couterexamples to addivity of output p-Rényi entropy (p>1) - a much simpler problem.
aubrun (arrobas) math. univ-lyon1. fr