Uniform entropy floors for optical sources with finite preparation memory

Can a light source use long correlations to evade the entropy generated by single-photon noise? This paper establishes a common noise floor for an entire family of sources with finite preparation resources, even when their memory mixes arbitrarily slowly. The source repeatedly applies the same isometry to a stationary quantum memory of dimension at most \(D\). Each emitted mode is supported on photon numbers \(0,\ldots,L\), with mean emission at least \(\mu>0\); the final preparation memory is traced out. There is no fresh discarded preparation noise. Every signal then meets an independent single-photon environment at a beam splitter. Write \(\Phi_{\eta,1}\) for the channel whose retained output equals the bath at \(\eta=0\). A source-independent interval, at every finite length. For fixed \(L,D,\mu\), there exist \(\eta_*(L,D,\mu)>0\) and \(v_*(L,\mu)>0\) such that every allowed source and every finite block length \(n\) satisfy \[\frac{1}{n}S\!\left(\Phi_{\eta,1}^{\otimes n}(\rho_n)\right)\ge h_2(\eta)+v_*\eta\log(1/\eta),\qquad 0<\eta<\eta_*.\] Here S is von Neumann entropy in nats and \(h_2(t)=-t\log t-(1-t)\log(1-t)\). The complementary bound holds near \(\eta=1\), replacing the surplus by \(v_*(1-\eta)\log[1/(1-\eta)]\). The statement allows singular stationary memory states, without an ergodicity, mixing-gap or minimum-eigenvalue assumption. Mechanism and usable scope. A dimension-paid entropy comparison retains the complete quantum memory while avoiding inverse conditioning. The persistent emitted field lies in the center of the reversible memory algebra. An explicit slow-memory family shows why recovering the full perturbative coefficient uniformly would be an unnecessarily strong requirement. The surplus coefficient is explicit, while the common endpoint width is proved to exist by compactness. The occupation cutoff and memory dimension are fixed resources; a mean-energy cap alone is a different source model. The integrated 11-page article contains two figure groups, complete proofs and references. It connects finite preparation resources to a uniform single-photon-noise entropy threshold; it does not assume a general arbitrary-input additivity theorem.