  The so-called stellar formalism allows 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
characterisation 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.
