# Stellar representation of non-Gaussian quantum states

## TL;DR

This paper introduces the stellar hierarchy, a novel way to classify non-Gaussian quantum states based on the zeros of their Husimi Q-function, with implications for quantum state engineering and computing.

## Contribution

It develops the stellar hierarchy, linking the zeros of the Husimi Q-function to state properties and minimal photon additions for state engineering.

## Key findings

- Defines the stellar hierarchy based on Husimi Q-function zeros
- Provides an operational method to engineer states in the hierarchy
- Analyzes topological properties of the hierarchy with respect to trace norm

## Abstract

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.