Abstract
We present analytical results toward the Wigner entropy conjecture, which posits that among all physical Wigner non-negative states the Wigner entropy is minimized by pure Gaussian states for which it attains the value 1+lnπ.Working under a minimal set of constraints on the Wigner function, namely, non-negativity, normalization, and the pointwise bound π W≤ 1, we construct an explicit hierarchy of lower bounds B_n on S[W] by combining a truncated series lower bound for -ln x with moment identities of the Wigner function.This yields closed-form purity-based sufficient conditions ensuring S[W]≥ 1+lnπ.In particular, we first prove that all Wigner non-negative states with μ≤ 4-2√3 satisfy the Wigner entropy conjecture. We further obtain a systematic purity-only relaxation of the hierarchy, yielding the simple sufficient condition μ≤ 2/e. On top of aforesaid results, our analysis clarifies why additional physicality constraints are necessary for purity-based approaches that aim to approach the extremal case μ≤1.