Detecting quantum non-Gaussianity via the Wigner function

TL;DR

This paper develops criteria based on the Wigner function to identify quantum non-Gaussian states of a harmonic oscillator, especially under noise, by establishing bounds that, when violated, confirm non-Gaussianity.

Contribution

The authors introduce new criteria using the Wigner function to detect quantum non-Gaussian states, including bounds applicable under noise and Gaussian operations.

Findings

Bound on Wigner function at the origin for Gaussian mixtures

Violation of the bound indicates non-Gaussianity

Criteria effective even for states with positive Wigner function under high losses

Abstract

We introduce a family of criteria to detect quantum non-Gaussian states of a harmonic oscillator, that is, quantum states that can not be expressed as a convex mixture of Gaussian states. In particular we prove that, for convex mixtures of Gaussian states, the value of the Wigner function at the origin of phase space is bounded from below by a non-zero positive quantity, which is a function only of the average number of excitations (photons) of the state. As a consequence, if this bound is violated then the quantum state must be quantum non-Gaussian. We show that this criterion can be further generalized by considering additional Gaussian operations on the state under examination. We then apply these criteria to various non-Gaussian states evolving in a noisy Gaussian channel, proving that the bounds are violated for high values of losses, and thus also for states characterized by a positive Wigner function.