A remark on the paper ``Randomizing quantum states: Constructions and applications''

TL;DR

This paper improves the efficiency of constructing $ ext{ extit{ε}}$-randomizing quantum channels by reducing the required number of random unitary operators from approximately $d ext{log} d$ to roughly $d$, simplifying previous methods.

Contribution

It introduces a simple trick that enhances the existing construction of $ ext{ extit{ε}}$-randomizing channels, decreasing the number of unitaries needed for approximate quantum state encryption.

Findings

Reduces the number of unitaries from $d ext{log} d$ to $d$

Simplifies the construction of $ ext{ extit{ε}}$-randomizing channels

Improves efficiency of quantum state encryption methods

Abstract

The concept of $\e$-randomizing quantum channels has been introduced by Hayden, Leung, Shor and Winter in connection with approximately encrypting quantum states. They proved using a discretization argument that sets of roughly $d \log d$ random unitary operators provide examples of such channels on $\C^d$. We show that a simple trick improves the efficiency of the argument and reduces the number of unitary operators to roughly $d$.