A novel non-Gaussianity measure based on the Wigner relative entropy*

The enhanced phase-space characteristics of non-Gaussian states of light, albeit necessary for universal quantum computing, render their understanding and production challenging. In attempts to circumvent these difficulties, several works have introduced non-Gaussianity measures, i.e., quantities that assign a real number to states depending on their non-Gaussian content (Genoni et al., 2007, 2008). Based on the Wigner entropy (Van Herstraeten & Cerf, 2021), we introduce a new measure μ_W(\{hat{ρ}), which is the Wigner relative entropy between an arbitrary N-mode state \hat{ρ} and its Gaussian associate \{hat{ρ}_G defined as 

μ_W(\hat{ρ}) = ∫ d^Nq d^Np W(q, p) [ln W(q, p)  - W_G(q, p)].

Here, W(q, p) and W_G(q, p) are the Wigner functions of the state and its Gaussian associate respectively. Our measure can be complex-valued, and we interpret its imaginary part as the negative volume of the Wigner quasi-distribution, while its real part provides information on other intrinsic properties of the state. We prove that μ_W(\hat{ρ}) is a valid non-Gaussianity measure and demonstrate its usefulness in representing states more perceptibly. In our work we discuss its relevance to non-Gaussian state generation and its connection to the more general context of resource theories.