We construct physical quantum states with everywhere positive Wigner functions whose Shannon entropy lies below the vacuum value 1+\ln\pi. Besides an explicit finite-energy counterexample, we obtain rank-two finite-Fock-support families with analyti
We construct physical quantum states with everywhere positive Wigner functions whose Shannon entropy lies below the vacuum value 1+\ln\pi. Besides an explicit finite-energy counterexample, we obtain rank-two finite-Fock-support families with analytic positivity and entropy bounds. For fixed Fock level n and coherence fraction 0\le\lambda<1, the entropy difference satisfies h(W)-(1+\ln\pi)=(2n-2^n\lambda^2)t^2+O_{n,\lambda}(t^4), yielding finite-support counterexamples for every n\ge3 above an explicit coherence threshold. We also show that the absence of a negative quadratic term does not preclude entropy descent: a fully coherent vacuum–one-photon core with a vanishing positive thermal repair gives h(W)-(1+\ln\pi)=-4t^6/3+o(t^6). For the vacuum–three-photon construction, we determine the logarithmic asymptotic of the minimum fixed-thermal mixing weight required for Wigner nonnegativity. Finally, optimizing over all one-mode Wigner-nonnegative states with mean photon number at most E, we prove that the maximal entropy deficit has the sharp scale E^\gamma/[\ln(1/E)]^\beta, where \gamma\simeq0.7412033679 and \beta\simeq0.5861054961. Thus the vacuum entropy is recovered as E\to0, but the optimal deficit decays much more slowly than any universal linear correction in the mean energy.