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Distillation of entanglement using only Gaussian operations is an importantprimitive in quantum communication, quantum repeater architectures, anddistributed quantum computing. Existing distillation protocols for continuousdegrees of freedom are only known to converge to a Gaussian state whenmeasurements yield precisely the vacuum outcome. In sharp contrast,non-Gaussian states can be deterministically converted into Gaussian stateswhile preserving their second moments, albeit by usually reducing their degreeof entanglement. In this work - based on a novel instance of a non-commutativecentral limit theorem - we introduce a picture general enough to encompass theknown protocols leading to Gaussian states, and new classes of protocolsincluding multipartite distillation. This gives the experimental option ofbalancing the merits of success probability against entanglement produced.

Gaussification and entanglement distillation of continuous variable systems: a unifying picture