Maximum output purity under correlated non-Gaussian optical noise

Can correlations in an optical environment allow an entangled signal to produce a purer output than coherent light? This article removes that input optimization for a specified class of non-Gaussian noise, providing an exact entropy benchmark for every joint signal state. Exact maximum over all inputs. Consider m signal modes coupled locally to an initially independent environment \[\omega_E=\sum_x p(x)|x\rangle\langle x|.\] The sum runs over all binary photon-number histories, with zero or one photon in each environment mode. Their probability law p may have arbitrary correlations across the entire block. Environmental outputs are discarded. For the transmissivity vector \(\vec{\eta}=(\eta_1,\ldots,\eta_m)\), the channel satisfies \[P_{\max}(p,\vec{\eta}):=\sup_{\rho}\operatorname{Tr}[\Phi_{p,\vec{\eta}}(\rho)^2]=\mathbb{E}_{X,Y\sim p}\prod_{j=1}^m h_{\eta_j}(X_j,Y_j),\] where \(X,Y\) are independent draws from the same complete bath law and \[h_\eta(0,0)=1,\qquad h_\eta(0,1)=h_\eta(1,0)=\eta,\qquad h_\eta(1,1)=\eta^2+(1-\eta)^2.\] The equality holds for every common \(\eta\in[0,1]\), and separately when all transmissivities lie in \([1/2,1]\) or all lie in \([0,7/24]\). The supremum includes arbitrary normal mixed and entangled inputs without a signal-energy restriction. Vacuum and every coherent product input attain it. The right-hand side is the collision probability of the vacuum-output distribution, not of the bath histories themselves. What correlations change. Two explicit baths with identical one- and two-mode marginals, identical corresponding reduced signal channels and equal mean photon budgets have exponentially versus polynomially decaying maximum output purity. At balanced coupling and 31 modes, their maxima differ by approximately 2080. The large separation already follows from a vacuum construction and the established independent-bath result; the new all-input theorem fixes the correlated bath's exact optimum. A Markov example turns the infinite-dimensional optimization into a four-dimensional transfer calculation. Mechanism and boundaries. The proof retains the even total relative-photon parity imposed by two identical signal copies and controls the cross terms between distinct bath histories. Convexity and bounded-operator continuity return the comparison to all normal inputs, including infinite photon tails. Diagonality of the bath is essential: an explicit coherent binary bath allows a noncoherent signal to outperform all coherent products. For arbitrary heterogeneous couplings uniformly separated from zero and one, exponential suppression of maximum purity is equivalent to an exponentially small probability of a whole-block low-occupation event. The results give lower bounds on output second Rényi entropy and necessary ranks for approximating the complete output. They do not determine ordinary-entropy additivity or communication capacity; a same-bath comparison shows that a history-conditioned entropy method can give a tighter communication bound than exact purity. The independent contribution extends exact all-input purity beyond independent non-Gaussian baths, while retaining the attribution of Gaussian extremality, replica methods and classical collision calculations. The integrated 19-page PDF contains two figure groups, complete proofs and references. Its finite zero-click example is a theoretical test of two specified bath models with explicit reset, detection and mean-energy assumptions.