The energy cost of optical noise

This work gives a resource law for engineering optical noise: how many photons must a single environment mode contain to prevent any signal from producing an output that is too spectrally concentrated? It provides an explicit optimal bath preparation, a matched Gaussian benchmark, and separately proved guarantees for correlated repeated use. Exact photon cost of a spectral noise guarantee A signal mixes with an independently prepared bath \(σ\) at a beam splitter of signal transmissivity \(η\). The requirement is \[Φη,σ(ρ)≤ cI every normal input state ρ, 0<c<1.\] Equivalently, every rank-one output projection has probability at most \(c\) and the receiver's output min-entropy is at least \(-ln c\). Inputs may be mixed, non-Gaussian, arbitrarily energetic, and have infinite photon tails. Let \(d= 1/c-1\). Throughout \(0<η≤ 1/ 1/c\), the minimum mean photon numbers of the actual bath are \[Nₐₗₗ(c,η)=d-cd(d+1)/2/1-η.\] \[NG(c,η)={c⁻¹-1}{1-η}.\] The first optimization includes all normal bath states and has a unique finite-support minimizer. The second includes all Gaussian density operators, with arbitrary squeezing and displacement; a centered thermal bath attains it. The coupling and spectral requirement are identical in both comparisons. At \(c=1/j\), for every integer \(j≥2\) and \(0<η≤1/j\), \[NG(1/j,η)=2Nₐₗₗ(1/j,η).\] Between reciprocal budgets the ratio varies. The cost counts mean photons in the physical bath, not laboratory preparation work or the energy of an auxiliary purification. A physical preparation and its complete single-use spectrum Writing \(r=1-dc\), the attaining vacuum-output spectrum is \[λ_*=({c,…,c}d entries,r).\] It majorizes every output of the same channel, so the minimum von Neumann entropy is exactly \[Sₘᵢₙ(Φη,σ)=-dcln c-rln r.\] The other positive-order Rényi minima are fixed by the same spectrum. Coherent inputs attain these minima. The proof identifies an actual optical dilation whose port antisymmetry enforces the all-input occupation bound. Exact examples also show that baths with identical energy and covariance can have different optimized entropy floors. What survives entangled repeated use For the same resource-optimal flat-output families of ranks \(j=2,3,4\), separate estimates prove \[Sₒᵤₜ≥ mln j\] for every finite number \(m\) of uses and arbitrary normal joint inputs, including cross-use entanglement. The environments vary with \(η\) along their specified families. Further results give sharp tensorized purity bounds for the entire three-level diagonal bath simplex and for every Fock environment at balanced coupling. The paper then treats a different, fixed bath, \(σ=11\), at every \(0<η<1\). With \[h(η)=-ηlnη-(1-η)ln(1-η),\] it proves: Exact \(Sₒᵤₜ≥ mh(η)\) at arbitrary finite blocklength for joint inputs invariant under each local photon-number phase rotation by \(2π/q\), \(q≥4\). Coherence and entanglement within residue sectors and infinite tails remain allowed. The residue-zero restricted channel also has strong minimum-output-entropy additivity with arbitrary normal partner channels. The same exact entropy floor for up to three globally total-phase-invariant input modes, without a mean-energy limit. \(Sₒᵤₜ≥ Mh(η)-o(M)\) for physical-mode permutation-invariant inputs of finite mean energy density, with explicit errors and full infinite-tail control. For every fixed \(m\) and interior \(η\), all outputs remain a uniformly positive trace distance from every state of rank below \(2^m\), without any input-energy bound. This covers nonbalanced parameters where the older purity-based distance bound vanishes; the new positive constant is spectrally implicit. Interpretation and scope The input classes above are not nested, and an entropy rate, a finite-block entropy floor and a low-rank distance are distinct guarantees. The fixed single-photon bath is not generally the resource-optimal bath. Complete unrestricted nonbalanced single-photon entropy additivity and exact classical capacity remain open. A spectral noise guarantee constrains output mixing; it does not mean that all input information or reference correlations are erased. The article separates this issue through actual resolved-erasure qubit restrictions, antidegradability of the specified optimal reciprocal families, and a classical capacity bracket under a true signal-energy budget. The integrated 52-page article contains six scientific figures, the complete proofs, and the mathematical data needed to reconstruct its finite certificates. Logarithms and entropy values use natural units unless bits are explicitly stated.