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.

**
E-mail : **

aubrun (arrobas) math. univ-lyon1. fr