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

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.