Upper bounds on the purity of Wigner positive quantum states that verify the Wigner entropy conjecture

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 $1+\ln \pi$.Working under a minimal set of constraints on the Wigner function, namely, non-negativity, normalization, and the pointwise bound $\pi W\le 1$, 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 $S[W]\ge 1+\ln \pi$.In particular, we first prove that all Wigner non-negative states with $\mu \le 4-2\sqrt{3}$ satisfy the Wigner entropy conjecture. We further obtain a systematic purity-only relaxation of the hierarchy, yielding the simple sufficient condition $\mu \le 2\mathrm{/}e$. 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 $\mu \le 1$.