No-broadcasting of non-Gaussian states

Gaussian states are of fundamental importance in the physics of continuous-variable quantum systems. They are appealing for the experimental ease with which they can be produced, and for their compact and an elegant mathematical description. Nevertheless, many proposed quantum technologies require us to go beyond the realm of Gaussian states and introduce non-Gaussian elements. In terms of quantum resource theory, we can then recognize non-Gaussian states as resources and Gaussian operations and states as free, which can be used and prepared easily. Given such a structure of resource theory, the task of broadcasting the resource is to determine if the resource content of a state can be cloned in a meaningful way, which, if possible, provides a strong operation for manipulation of the resource. In this work, we prove that broadcasting of non-Gaussian states via Gaussian operations is not possible. For this, we first show that the relative entropy of non- Gaussianity is not super-additive, which rules it out as a prime candidate in the analysis of such no-go results. Our proof is then based on understanding fixed points of Gaussian operations and relates to the theory of control systems. The no-go theorem also states that if two initially uncorrelated systems interact by Gaussian dynamics and non-Gaussianity is created at one subsystem, then the non-Gaussianity of the other subsystem must be reduced. Further, keeping the set of free operations fixed to Gaussian operations, we can also comment on the broadcasting of Wigner negativity and genuine quantum non-Gaussianity.