Stellar Representation of Non-Gaussian Quantum States

The so-called stellar formalism allows us to represent the non-Gaussian properties of single-mode quantum states by the distribution of the zeros of their Husimi Q function in phase space. We use this representation in order to derive an infinite hierarchy of single-mode states based on the number of zeros of the Husimi Q function: the stellar hierarchy. We give an operational characterization of the states in this hierarchy with the minimal number of single-photon additions needed to engineer them, and derive equivalence classes under Gaussian unitary operations. We study in detail the topological properties of this hierarchy with respect to the trace norm, and discuss implications for non-Gaussian state engineering, and continuous variable quantum computing.