Before writing this paper, I posted on arXiv a short note intituled "A remark on the paper "Randomizing quantum states: constructions and application" ". The new paper improves the result with a minor variation of the argument ; the short note will not be published.

We study quantum channels on C^d obtained by
selecting randomly N independent Kraus operators according to a
probability measure on the unitary group. Such channels are "almost
randomizing" when N is large, the problem is to quantify the speed of
convergence.
This problem has similarities with the problem studied in paper
[5]
of the present list.

For the Haar measure, we slightly improve on results by Hayden, Leung,
Shor and Winter by showing that N can be taken proportional to d. For
other measures (such as the uniform measure on products of Pauli
matrices),
we show that N can be taken propotional to d(log d)^6. The proof uses
recent results on empirical processes in Banach spaces.

For people who like killing logarithms: the result could probably be improved to d(log d).

Here are two Beamer presentations of the paper :

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E-mail : **

aubrun (arrobas) math. univ-lyon1. fr