Sharp entropy bounds from thermal local records and photon fluctuations

When an optical source reproduces the local records of a thermal state, how much microscopic mixedness do those records actually guarantee? This paper gives explicit entropy bounds and a sharp large-budget law for that inference when the source's photon fluctuations are bounded. The setting is one oscillator with \([Q,P]=i\) and \(N=(Q^2+P^2-1)/2\). Fix a mean photon number \(E>0\), and put \(v=E+1/2\) and \(A=v-1/(4v)\). The complete position density, probability current and local kinetic density are required to be \[n_v(q)=\frac{e^{-q^2/(2v)}}{\sqrt{2\pi v}},\qquad j_v(q)=0,\qquad k_v(q)=\left(\frac{q^2}{4v^2}+A\right)n_v(q).\] These are exact functions, with no factor \(1/2\) in the definition of \(k_v\). They force mean photons \(E\). The additional resource is the actual expectation \(\operatorname{Tr}(N^2\rho)\le M\) of the same source. At fixed mean, this is also a photon-number-variance cap. For every feasible finite \(M\), every normal state compatible with these records satisfies \[S(\rho)\ge h_2\!\left(\min\{\eta_E(M),1/2\}\right)>0.\] \[\eta_E(M)=\left[\frac{A}{\sqrt M+\sqrt{M+A(A+B)}}\right]^2,\qquad B=1-\frac1{2v}.\] Here \(S(\rho)=-\operatorname{Tr}(\rho\log\rho)\), \(h_2(t)=-t\log t-(1-t)\log(1-t)\), and logarithms are natural. The bound applies to all compatible normal states, with no Gaussian or finite-rank restriction. Let \(S_{\min}(E,M)\) be the entropy infimum over that full compatible class. As \(M\to\infty\) at fixed \(E>0\), \[S_{\min}(E,M)=\frac{c(E)\log M+c(E)[1-\log c(E)]}{M}+O_E\!\left(\frac{\log M}{M^2}\right).\] The coefficient is \(c(E)=E^2(E+1)^2/(2E+1)^2\). A positive mixture of two pure Gaussian states preserves all three complete records and has an exactly computed photon-square cost. It supplies the matching upper bound, fixing both the leading coefficient and the next term. The same asymptotic law therefore holds within actual probability mixtures of pure Gaussian states. Finite-budget results are explicit bounds; the sharp equality is the stated fixed-\(E\), large-\(M\) expansion. Without a photon-square cap, the exact compatible entropy range is \((0,g(E)]\), where \(g(E)=(E+1)\log(E+1)-E\log E\). The thermal state uniquely attains the upper endpoint, while compatible Gaussian mixtures approach zero entropy. This separates a maximum-entropy model from the minimum entropy forced by the data. The proof uses established local Gram relations, projects onto an actual largest spectral component, and charges its positive spectral tail to the same source's \(N^2\) budget. This connection retains the column correlations needed for the lower bound. The analysis concerns one mode and ideal complete records; finite-precision or sampled records require additional error control. The integrated 13-page PDF contains two figures, complete proofs and references.