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 The new paper improves the result with a minor variation of the argument ; the short note will not be published.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
This problem has similarities with the problem studied in paper
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 :
aubrun (arrobas) math. univ-lyon1. fr