Discretization of quantum pure states and local random unitary channel

TL;DR

This paper demonstrates that a quantum channel formed by averaging a logarithmic number of random unitaries can effectively randomize quantum pure states locally, using an epsilon-net approach and concentration analysis.

Contribution

It introduces a method to construct local epsilon-randomizing quantum channels with a small number of unitaries based on epsilon-net techniques.

Findings

A small epsilon-net on the quantum state space enables effective analysis.

A logarithmic number of random unitaries suffices for local epsilon-randomization.

Such channels are shown to exist with non-zero probability.

Abstract

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.