How concentrated can a photon-number distribution be without Wigner negativity?

A sharply concentrated photon-number distribution is a basic target in the preparation of quantum light. We determine the strongest concentration compatible with a nonnegative Wigner function. As the photon number \(n\) grows, its maximal occupation probability obeys \[ω_n:=ρ:\,W_ρ≥0 n|ρ|n=Θ(n-1/3).\] The upper bound covers all Wigner-positive quantum states, while displaced squeezed Gaussian states attain the same order. Within the same class, the minimum photon-number variance at large mean photon number \(μ\) is \(Θ(μ2/3)\). The concentration law also fixes a noise cost for removing the Wigner negativity of an \(n\)-photon state. Let \(R_g\) be the least noise-to-signal mixing ratio when any quantum noise state is allowed, and \(R_s\) the corresponding ratio when the added noise must itself be Wigner-positive. We obtain \[{aligned}1+R_g(|n n|)&=ω_n⁻¹=Θ(n1/3),\\1+R_s(|n n|)&=Θ(n1/2).{aligned}\] The allowed noise therefore changes the asymptotic cost of erasing the same negative Wigner function. Certified thresholds at finite photon number turn these limits into photon-counting tests of Wigner negativity.