Random purification channel made simple

TL;DR

This paper presents a simple construction of the random purification channel, demonstrating its ability to purify non-i.i.d. states and providing a concise proof of a strengthened Uhlmann's theorem for quantum divergences.

Contribution

The paper introduces a straightforward construction of the random purification channel, extending its applicability to non-i.i.d. states and simplifying proofs of key quantum information theorems.

Findings

The channel can purify permutationally symmetric states into convex combinations of symmetric purifications.

It provides a simple, transparent way to understand the properties of the random purification channel.

A one-line proof of a stronger version of Uhlmann's theorem for quantum divergences is established.

Abstract

The recently introduced random purification channel, which converts $n$ i.i.d. copies of any mixed quantum state into a uniform convex combination of $n$ i.i.d. copies of its purifications, has proved to be an extremely useful tool in quantum learning theory. Here we give a remarkably simple construction of this channel, making its known properties -- and several new ones -- immediately transparent. In particular, we show that the channel also purifies non-i.i.d. states: it transforms any permutationally symmetric state into a uniform convex combination of permutationally symmetric purifications, each differing only by a tensor-product unitary acting on the purifying system. We then apply the channel to give a one-line proof of (a stronger version of) the recently established Uhlmann's theorem for quantum divergences.