Random purification channel for passive Gaussian bosons

TL;DR

This paper introduces a Gaussian version of the random purification channel for passive Gaussian bosonic states, enabling the generation of purifications with doubled mean photon number using advanced representation theory.

Contribution

It constructs a Gaussian random purification channel for passive bosonic states, with each purification having twice the mean photon number, based on the representation theory of passive Gaussian unitaries.

Findings

Constructed a Gaussian random purification channel for passive bosonic states.

Each purification has exactly twice the mean photon number of the original state.

Utilized representation theory of dual reductive pairs of unitary groups.

Abstract

The random purification channel, which, given $n$ copies of an unknown mixed state $\rho$, prepares $n$ copies of an associated random purification, has proved to be an extremely valuable tool in quantum information theory. In this work, we construct a Gaussian version of this channel that, given $n$ copies of a bosonic passive Gaussian state, prepares $n$ copies of one of its randomly chosen Gaussian purifications. The construction has the additional advantage that each purification has a mean photon number which is exactly twice that of the initial state. Our construction relies on the characterisation of the commutant of passive Gaussian unitaries via the representation theory of dual reductive pairs of unitary groups.