The active entropy photon-number inequality: exact Gaussian-channel minima and rigidity of near-optimal inputs

Abstract How much does a nearly minimal output entropy reveal about the quantum states entering an amplifier? We prove the active entropy photon-number inequality for two independent multimode inputs, each with arbitrary internal correlations, and characterise every equality case. Writing 𝒩n(ρ) = g−1(S(ρ)/n), where g(x) = (x + 1) log(x + 1) − x log x, the output obeys 𝒩n(C) ≥ G𝒩n(A) + (G − 1)[𝒩n(B) + 1]. Equality requires Gaussian inputs with one common symplectic shape, reflected on the creation-operator port. At fixed finite mode number and input energy, a power of the actual output gap bounds the full trace distance to a compatible Gaussian pair with the original means and entropies. For a pure one-mode signal and a specified Gaussian idler, a Wehrl-entropy bridge gives a square-root bound. The inequality also fixes the exact constrained output-entropy minimum of every finite block of thermal amplifiers. For unequal one-mode Gaussian channels, the optimisation over all entangled and non-Gaussian block inputs reduces to scalar entropy allocation, with a sharp threshold at which a second mode becomes active. These results determine both the optimal output and the input structure that can approach it, turning an entropy benchmark into a quantitative test of source models. Version 1.2 — 8 October 2026 This revision develops the input-rigidity theory: near saturation of the active EPnI now constrains both independent inputs jointly. Complete equality classification. For two independent finite-energy multimode inputs, equality is characterised by a common symplectic Gaussian shape, with reflection on the creation-operator port. Each input may contain arbitrary internal correlations. Quantitative stability for arbitrary inputs. At fixed finite mode number, gain and input-energy bound, a power-law estimate controls the full trace distance to one compatible Gaussian pair matching the original means and entropies. The theorem includes pure and nonfaithful states and states with infinite Fock support. A stronger one-mode bound. For a pure signal and a specified Gaussian idler, which may be mixed, a Wehrl-entropy bridge gives a square-root estimate and an explicit sufficient tolerance for the input error. Expanded proofs and figures. New appendices provide the complete arguments and explicit constants. A new vector figure connects the equality geometry with the quantitative input-error bound. The active EPnI, exact finite-block thermal-amplifier minima and exact scalar entropy allocation for heterogeneous Gaussian channels are retained. The 49-page PDF contains the complete manuscript, proofs, four figures and references.