Physical Counterexamples to the Wigner Shannon Entropy Conjecture

• URL: http://arxiv.org/abs/2609.13312v1
• Date: Thu, 10 Sep 2026 15:55:43 GMT
• Title: Physical Counterexamples to the Wigner Shannon Entropy Conjecture
• Abstract summary: We construct quantum states with everywhere positive Wigner functions whose Shannon entropy lies below the vacuum value $1+ln$.<n>For fixed Fock level $n$, the entropy difference satisfies $h(W)-(1+ln)= (2n-2n2)t2+O_n,(t4)$.<n>For the vacuum--three-photon construction, we determine the logarithmic of the minimum fixed-thermal mixing weight required for Wigner nonnegativity.
• Score: 1.5873881835353227
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We construct physical quantum states with everywhere positive Wigner functions whose Shannon entropy lies below the vacuum value $1+\lnπ$. 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 $0\leλ<1$, the entropy difference satisfies $h(W)-(1+\lnπ)=(2n-2^nλ^2)t^2+O_{n,λ}(t^4)$, 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 $h(W)-(1+\lnπ)=-4t^6/3+o(t^6)$. 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 $E^γ/[\ln(1/E)]^β$, where $γ\simeq0.7412033679$ and $β\simeq0.5861054961$. 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.

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