A random purification channel for arbitrary symmetries with applications to fermions and bosons

TL;DR

This paper introduces a generalized random purification channel for states with arbitrary symmetries, leading to new fermionic and bosonic Gaussian purification methods and efficient tomography protocols.

Contribution

It extends the random purification channel concept to arbitrary symmetry groups, providing new tools for fermionic and bosonic quantum state purification.

Findings

Constructed a quantum channel for states in algebra generated by any symmetry group G.

Proved a concise version of the original random purification theorem.

Developed an optimal tomography protocol for fermionic Gaussian states.

Abstract

The random purification channel maps n copies of any mixed quantum state to n copies of a random purification of the state. We generalize this construction to arbitrary symmetries: for any group G of unitaries, we construct a quantum channel that maps states contained in the algebra generated by G to random purifications obtained by twirling over G. In addition to giving a surprisingly concise proof of the original random purification theorem, our result implies the existence of fermionic and bosonic Gaussian purification channels. As applications, we obtain the first tomography protocol for fermionic Gaussian states that scales optimally with the number of modes and the error, as well as an improved property test for this class of states.