Gaussianity bounds for quantum mixed states with a positive Wigner function

Abstract

According to Hudson's theorem, the only quantum pure states with a positive Wigner function are Gaussian states. We summarize and compare some recent attempts at extending this theorem to the space of quantum mixed states, and complement them with new results obtained from numerical observations. Specifically, we look for upper bounds on the admissible non-Gaussianity of mixed states with a positive Wigner function.