Random purification channel for general (non‑passive) Gaussian states

Construct a random purification channel that, for any n‑mode bosonic Gaussian state, maps N copies of the state to N copies of a randomly chosen Gaussian purification whose covariance operator norm is bounded polynomially in that of the original state.

Background

The authors introduce a passive random purification channel that enables sample‑optimal tomography for passive Gaussian states by converting multiple copies of a mixed state into copies of a random pure Gaussian purification.

They point out that extending this construction beyond the passive case is technically challenging due to the non‑compactness of the symplectic group (squeezing), but would yield sample‑optimal learning for all Gaussian states via their reduction.

References

One open problem in this direction is how to construct a random purification channel for general (non-passive) Gaussian states. The main technical challenge there dealing with the non-compactness of the symplectic group, which is necessary to describe squeezing. Though solving this problem is left for future works, we provide the following reduction result, which follows from the same reasoning as in the proof of Theorem~\ref{th:main_up_passive}.

— Towards sample-optimal learning of bosonic Gaussian quantum states  (2603.18136 - Chen et al., 18 Mar 2026) in Upper bounds on Gaussian state learning, after Theorem 4 ("Non-Gaussian advantage in learning passive Gaussian states")