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 $1+\ln\pi$. 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 sufficient conditions, expressed in terms of the state purity $\mu:=\operatorname{Tr}(\rho^2)=2\pi\int dq\,dp\,W(q,p)^2$, ensuring $S[W]\ge 1+\ln\pi$. 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.