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 $S\left[W\right]$ by combining a truncated series lower bound for $-lnx$ with moment identities of the Wigner function. This yields closed-form sufficient conditions, expressed in terms of the state purity $\mu :=Tr\left({\rho }^2\right)=2\pi \int dqdpW{(q,p)}^2$, ensuring $S\left[W\right]\ge 1+ln\pi$. In particular, we first prove that all physical 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, whose limiting sufficient condition is $\mu \le 2/e$. Finally, we show that the threshold $2/e$ is sharp under the relaxed constraints considered here, thereby identifying the need for additional quantum-realizability information in the remaining high-purity regime.

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• Quantum information theory