Coherence and sharp endpoint laws for photon counting in quantum amplifiers

How far can an amplifier's complete photon-count record depart from its coherent-light reference when the output entropy and the source's mean-photon budget are known? This article determines sharp endpoint laws for that inference. Photon counting reads occupations, but coherence changes the entropy cost of producing those occupations; the available energy budget selects the most efficient coherent pattern. A common physical comparison. For one optical mode, let \(\mu=\mathrm{Tr}(a\rho)\), \(\rho_c=D(-\mu)\rho D(\mu)\), and \(E_c=\mathrm{Tr}(N\rho)-|\mu|^2\). The reference is the source's actual field mean. For the quantum-limited amplifier \(\mathcal A_2\) of gain two with a vacuum auxiliary mode, define \[\Delta(\rho)=g^{-1}(S(\mathcal A_2\rho))-1,\qquad T(\rho)=\tfrac12\sum_{k\ge0}\left|\langle k|\mathcal A_2(\rho_c)|k\rangle-2^{-k-1}\right|,\] where \(S\) is von Neumann entropy in nats and \(g(x)=(x+1)\log(x+1)-x\log x\). Thus \(\Delta\) is an output quantum-entropy excess expressed in photon units, rather than the Shannon entropy of the observed counts. The constrained frontier is \[F_E(\delta)=\sup\{T(\rho):E_c(\rho)\le E,\ \Delta(\rho)\le\delta\}.\] Three sharp endpoints. For every fixed finite \(E>0\), \[F_E(\delta)=\frac9{16}\delta+o(\delta),\qquad\delta\downarrow0.\] Weak squeezing attains this coefficient. The corresponding local ratio also holds on every fixed finite number of modes, using the complete joint count record and the total entropy photon-number excess. Its neighbourhood may depend on the fixed mode count and energy budget. For centred mean-photon budgets \(m=1,2\), put \(\Lambda_m=1-2^{-m}\) and \(\delta_m^{\mathrm{sat}}=g^{-1}(S(\mathcal A_2(|m\rangle\langle m|)))-1\). The exact saturated frontier is \(F_m(\delta)=\Lambda_m\) for \(\delta\ge\delta_m^{\mathrm{sat}}\), and its approach from below obeys \[F_m(\delta)=\Lambda_m-c_m(\delta_m^{\mathrm{sat}}-\delta)+o(\delta_m^{\mathrm{sat}}-\delta),\] with \(c_1\simeq0.73151177\), \(c_2\simeq0.16207193\), \(\delta_1^{\mathrm{sat}}\simeq0.92689630\), and \(\delta_2^{\mathrm{sat}}\simeq1.70174873\). The article gives exact expressions and certified numerical enclosures. Mean-cancelling vacuum/one-/two-photon superpositions attain the first saturation slope; phase-aligned zero-/two-/four-photon superpositions attain the second. The pure-state and full mixed-state optimizations have the same endpoint coefficients. Scope and mechanism. The budgets constrain centred mean energy, not the maximum photon number in each preparation. The proofs retain arbitrary normal mixed states and moving infinite photon tails. They combine uniform entropy comparisons with a decomposition that keeps the coherent part and mixed remainder tied to the same physical source. The endpoint laws concern a fixed ideal channel and its full count distribution; learning the reference and estimating entropy require separate measurement resources. The intermediate frontiers are not interpolated or claimed to be globally solved. This independent article complements entropy-minimization and single-event no-click bounds by treating the complete count law with actual-mean compensation. The integrated 27-page PDF contains four figure groups, complete proofs, reconstructible coefficient certificates and references.