Random purification channel for passive Gaussian bosons
Francesco Anna Mele, Filippo Girardi, Senrui Chen, Marco Fanizza, Ludovico Lami
TL;DR
The paper extends the random purification channel to bosonic passive Gaussian states, constructing a Gaussian purification map that outputs i.i.d. random Gaussian purifications whose mean photon number is exactly twice that of the input. The construction relies on a detailed commutant analysis of passive Gaussian unitaries via Howe duality, producing a strongly convergent channel Λ^{(n)} that uses Haar-averaged, k-filtered maximally entangled resources. This yields a practical pathway for Gaussian-state tomography and learning by reducing mixed-state tomography to purifications of Gaussian states, with controlled energy overhead. The work also clarifies the structure of the passive Gaussian commutant and connects to related efforts on energy-preserving Gaussian transformations.
Abstract
The random purification channel, which, given $n$ copies of an unknown mixed state $ρ$, 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.