No-Broadcasting of Non-Gaussian States

Gaussian states are central to continuous-variable quantum systems, as they are experimentally accessible and admit a compact mathematical description. Nevertheless, many proposed quantum technologies require non-Gaussian elements. In resource-theoretic terms, non-Gaussian states are resources, while Gaussian states and operations are free. The task of broadcasting, then, asks whether the resource content of a state can be meaningfully cloned. In this Letter, we prove that broadcasting of non-Gaussian states via Gaussian operations is not possible. We first show that the relative entropy of non-Gaussianity is not superadditive, ruling it out for such no-go analyses. Our proof instead uses fixed points of Gaussian operations and relates to control theory. The theorem also implies that if two initially uncorrelated systems interact by Gaussian dynamics, and one subsystem gains non-Gaussianity, the other must lose it irreversibly. Keeping the set of free operations fixed to Gaussian operations, we also comment on the broadcasting of Wigner negativity and genuine quantum non-Gaussianity.