Hidden quantum coherence at the Wigner entropy endpoint

How much high-photon quantum coherence is controlled by a nonnegative phase-space distribution? This article gives quantitative guarantees for approximating an optical state by a finite photon-number block, and constructs physical states showing why the guarantee cannot be inferred from mean energy or Wigner positivity alone at the strength available for a Gaussian-mixture source. Source classes and scope. The comparison concerns one optical mode with \([Q,P]=i\), mean photon number at most \(E>0\), and a known parity centre. The full class \( W_E\) consists of normal Wigner-positive states at the exact vacuum min-entropy endpoint, equivalently states supported on the even photon sector in that frame. The subclass \( G_E\) consists of actual probability mixtures of centred pure squeezed vacua. Mixture labels are unavailable, component energies may be arbitrarily large, and their true average energy is charged. Infinite photon tails are retained. A hierarchy of coherent-tail guarantees. Let \(P_L\) project onto photon numbers at most \(L\), and define the cutoff error by \[e_L(ρ)=12\|ρ-P_Lρ P_L\|_1.\] This quantity includes both discarded photon population and coherence across the cutoff. A mean-energy constraint alone gives the standard \(O_E(L-1/2)\) guarantee. At the exact Wigner endpoint, the article proves \[ρ∈ W_Ee_L(ρ)≤ C_E(L+1)-3/4, ρ∈ G_Ee_L(ρ)≤ 4E/L+1, L≥2.\] The stronger endpoint bound combines operator positivity and Wigner positivity on the same density kernel, followed by a controlled conversion to trace norm. It requires only mean energy, without a uniform second photon moment. Physical coherence beyond the Gaussian-mixture bound. A family \(τ_d∈ W_E\) has exact energy \(E\), covariance \((E+1/2)I\), strictly positive Wigner functions and nonzero population at every even photon number. For cutoffs \(L_d=Θ(d^3)\), \[eL_d(τ_d)≥ c_Ed-5/2, (L_d+1)eL_d(τ_d)∞.\] Thus the Gaussian-mixture \(L⁻¹\) law fails uniformly on the full endpoint class. The construction preserves coherence through an actual optical partial trace and pays its physical energy cost. Positive phase mixing further yields distinct states with identical complete photon statistics and covariance but nearly indistinguishable bounded optical responses. The rare bright branches obey the prescribed mean-energy budget; no efficient experimental preparation protocol or uniform branch-energy bound is assumed. Consequences for a specified measurement system. Each independent copy is measured by bounded centred squeezing followed by a vacuum/nonvacuum binary test. At fixed energy, squeezing range and confidence, let \(n_X(δ)\) be the minimum worst-case number of copies for trace-distance accuracy \(δ\) over source class \(X\). As \(δ↓0\), the bounds are \[log nG_E(δ)=Θ(δ⁻¹), cδ-6/5≤log nW_E(δ)≤ Cδ-4/3.\] These are measurement-restricted reconstruction costs. The lower bounds allow adaptive randomized settings. The upper bounds use finite designs and estimates in the original source class, with the energy budget respected for every record; they establish statistical existence rather than an efficient catalogue-generation algorithm. The full-class exponents remain distinct lower and upper bounds. The article also characterizes infinite-order Wigner entropy equality by displaced even-parity support and corrects the pure-Gaussian-only equality clause at infinite order in Dias and Prata (2023). The 27-page integrated article includes the complete proofs, figures and references.