Discretization of quantum pure states and local random unitary channel

We show that a quantum channel $\mathcal{N}$ constructed by averaging over $\mathcal{O}(\log d/\epsilon^2)$ randomly chosen unitaries gives a local $\epsilon$-randomizing map with non-negative probability. The idea comes from a small $\epsilon$-net construction on the higher dimensional unit sphere or quantum pure states. By exploiting the net, we analyze the concentrative phenomenon of an output reduced density matrix of the channel, and this analysis imply that there exists a local random unitary channel, with relatively small unitaries, generically.