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

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 $πW\le 1$, 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 $S[W]\ge 1+\lnπ$.In particular, we first prove that all Wigner non-negative states with $μ\le 4-2\sqrt3$ satisfy the Wigner entropy conjecture. We further obtain a systematic purity-only relaxation of the hierarchy, yielding the simple sufficient condition $μ\le 2/e$. 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 $μ\leq1$.