Physical Counterexamples to the Wigner Shannon Entropy Conjecture

Abstract

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.