Sharp entropy thresholds in multimode coherent-state approximation

When does finite phase-space entropy stop guaranteeing proximity to coherent light? This article determines sharp thresholds for two broad optical source classes when the number of modes may grow. It separates the roles of entropy, mode number and photon energy in approximating an optical state by a single pure coherent state. For an \(m\)-mode state \(ρ\), write \(Q_ρ(α)=α|ρ|α\), with \(dμ_m=π⁻ᵐd²ᵐα\), and define \[M(ρ)=2^m∫ Q_ρ(α)^2\,dμ_m(α), s=-ln M(ρ), D(ρ)=inf_α\|ρ-|αα|\|_1.\] Here \(s\) is the second-order Wehrl entropy excess and \(D\) is the full trace-norm distance, whose maximum is 2. The collision quantity \(M\) is also the output purity after half loss and can be accessed using interference of two independent, identically prepared copies. Exact source-class thresholds. The two classes are states diagonal in a common passive-mode number basis, and arbitrary states supported on the vacuum together with configurations whose photons occupy only one mode at a time in a common passive basis. The latter class retains arbitrary complex coherences, including vacuum coherences. For either class \( X\), with \(FX(s)=ρ∈ X,\,-ln M(ρ)=sD(ρ)^2\) and the supremum over all finite mode numbers, \[FX(s)=4 s≥ln4.\] The plateau is approached through growing-mode sequences, not attained by a finite-mode state. At its onset, vanishing coherent overlap and \(M→1/4\) force the one-photon probability to approach one, without an energy-bound assumption. This probability rigidity does not assert convergence of the mean energy. Sharp boundary geometry and energy. For the full number-diagonal class \( D\), the approach from below has the exact corner \[lims↑ln4{4-FD(s)}{ln4-s}=4{√ e}.\] A small vacuum component changes the optimal coherent peak and produces the sharp slope. Separate all-state energy bounds show that vanishing coherent overlap requires at least one photon asymptotically, or two photons for pure states. Explicit sources attain these limiting energy costs, and exact-entropy constructions include their additional photon costs. The proofs retain arbitrary photon tails and constants uniform in the mode number. The earliest-threshold and corner results apply to the source classes stated above. The integrated article contains the complete proofs and two figure groups.