A random purification channel for arbitrary symmetries with applications to fermions and bosons
Michael Walter, Freek Witteveen
TL;DR
The paper generalizes the random purification channel to arbitrary symmetry algebras by formulating a twirl-based construction over symmetry groups, with outputs restricted to the corresponding symmetric subspace. It recasts the original channel as a special case and provides a unified framework that yields new Gaussian-purification channels for both fermionic and bosonic systems. For fermionic Gaussian states, the authors derive a sample-optimal tomography and testing protocol with quadratic scaling in the number of modes, and prove a matching lower bound, establishing optimality. For bosonic Gaussian states, they establish a gauge-invariant purification channel by block-decomposing by particle number and twirling over passive Gaussian unitaries, indicating potential tomography advances in continuous-variable settings. Overall, the work leverages representation theory and algebraic twirling to connect purification, tomography, and Gaussian-state analysis in a broadly applicable framework.
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.