How much detected thermal light is needed to identify several emitting sites? This working paper studies binary sources on a known grid of addressable source-image modes, using one fixed passive optical mask, photon counting, and a predetermined observation length. It distinguishes the original one-reference apparatus, complete Fourier reference collection, and a separate apparatus with one added vacuum input. All recorded photons count toward the measurement cost. The central result establishes the optimal quadratic detected-photon order at unit brightness for every fixed support limit on every grid, without geometric restrictions or primality assumptions. A reciprocal transmission mask attains this order using target counts alone, while recorded reference photons remain included in the cost. The proof reconstructs short cyclic arcs from two rational-probe values, then uses contour integration to isolate entire clusters of poles. This gives uniform quantitative separation without requiring a fixed minimum distance between individual poles. A bounded candidate list removes the grid-size logarithm from the final comparisons. On prime grids, a shorter finite-field argument recovers the support from two integer edge profiles. The quadratic law also holds uniformly over every nonempty reference subset when known brightness remains in a fixed interval bounded away from zero. The contour argument extends to a general stability theorem for rational interaction observables: under explicit locality and conditioning assumptions, quantitative distinguishability within clusters yields quantitative separation of the global polynomial data. For irregular registered dictionaries, this gives a separation margin proportional to the minimum site spacing, with an optimal spacing exponent. This extension assumes exactly resolved distance channels; their optical acquisition and robustness to unknown position errors require further analysis. Explicit dependence on the support limit also yields a growing-support regime: the photon cost remains \(n^{2+o(1)}\log(1/\delta)\) whenever \(s^{2}\log(2s)=o(\log n)\). The accompanying computational bound is stated in real arithmetic, and the dependence on sparsity is deliberately conservative. The manuscript retains explicit ramp-mask constructions for every configuration of at most four emitters, for short-span sources, and for sources with distinct cyclic pair distances. The last class has no span restriction and contains an asymptotically overwhelming fraction of fixed-size supports. For four emitters, ordinary-correlation rigidity and cyclic separation together distinguish reflected configurations. An explicit passive mask also attains the exact optimal Fourier-intensity separation under a prescribed transmission floor. In both the original and complete-reference experiments, sufficiently faint light produces a different law: at fixed sparsity, the optimal photon order grows as the grid size raised to twice the support limit. For at most two emitters, the paper determines the optimal detected-photon order throughout the known-brightness range, from arbitrarily faint light through unit brightness, on odd grids in the original apparatus and on every grid with complete reference collection. The result includes the dependence on grid size, brightness, and error probability, including an additional logarithmic information cost between the faint and bright limits. At fixed confidence, the quadratic regime begins at brightness of order \((\log n)/n^{2}\). A mask combining an index ramp with a quadratic-residue code attains the sharp order. On even physical grids, one known vacuum input and odd-dimensional Fourier optics give the same law; omitting one target output preserves the detector count. This is a distinct apparatus. For the specified two-scale transmissions on an infinite even-grid family, an exact lower bound using the complete count record proves that changing phases or decoders alone cannot remove their intermediate-brightness limitation. A collected-light model makes the physical scope explicit. For a known collection operator, binary-source identifiability is equivalent to injectivity of the induced thermal covariance map. Covariance collisions give identical quantum states and therefore prevent discrimination by any subsequent receiver. A passive multiport establishes population identifiability when the map is injective. The paper also derives the exact throughput for uniform passive mode separation from the smallest Gram eigenvalue and accounts separately for detected and incident photons. These results connect the assumed mode access to a specified collection model; they do not establish a general quantum-optimal receiver or a complete astronomical implementation. Reference excess variance supplies a normalization that identifies the spatial source despite an unknown finite common-profile thermal-mode spectrum and independent unknown Poisson backgrounds. The finite-sampling guarantee includes the empty source under explicit mean and nonzero-signal bounds. Population identifiability of the mode spectrum is distinguished from stable estimation. When background means are known, the arm means recover source mass and improve a guaranteed scale-free precision radius. Further results include exact thermal count variances, confidence exponents, calibration bounds, a photon penalty for discarding detector labels, and an exact target-only localization obstruction when singleton brightness is unrestricted and unknown. The results have analytical proofs, supported by 34 reproducible finite and symbolic verification suites. These checks provide corroboration, not experimental validation or external peer review. The unrestricted fixed-sparsity unit-brightness problem is resolved on all grids. Remaining questions include the optimized even-grid intermediate-brightness transition in the original apparatus, transitions for three or more emitters, sharp growing-support rates, sharp reference-coverage dependence at vanishing brightness, and optimal covariance-only scale-free precision. Continuous off-grid recovery requires a tolerance-based criterion and additional stability analysis. The work makes no literature-priority claim.
How much detected thermal light is needed to identify several emitting sites? This working paper studies binary sources on a known grid of addressable source-image modes, using one fixed passive optical mask, photon counting, and a predetermined observation length. It distinguishes the original one-reference apparatus, complete Fourier reference collection, and a separate apparatus with one added vacuum input. All recorded photons count toward the measurement cost. The central result establishes the optimal quadratic detected-photon order at unit brightness for every fixed support limit on every grid, without geometric restrictions or primality assumptions. A reciprocal transmission mask attains this order using target counts alone, while recorded reference photons remain included in the cost. The proof reconstructs short cyclic arcs from two rational-probe values, then uses contour integration to isolate entire clusters of poles. This gives uniform quantitative separation without requiring a fixed minimum distance between individual poles. A bounded candidate list removes the grid-size logarithm from the final comparisons. On prime grids, a shorter finite-field argument recovers the support from two integer edge profiles. The quadratic law also holds uniformly over every nonempty reference subset when known brightness remains in a fixed interval bounded away from zero. The contour argument extends to a general stability theorem for rational interaction observables: under explicit locality and conditioning assumptions, quantitative distinguishability within clusters yields quantitative separation of the global polynomial data. For irregular registered dictionaries, this gives a separation margin proportional to the minimum site spacing, with an optimal spacing exponent. This extension assumes exactly resolved distance channels; their optical acquisition and robustness to unknown position errors require further analysis. Explicit dependence on the support limit also yields a growing-support regime: the photon cost remains (n^{2+o(1)}\log(1/\delta)) whenever (s^{2}\log(2s)=o(\log n)). The accompanying computational bound is stated in real arithmetic, and the dependence on sparsity is deliberately conservative. The manuscript retains explicit ramp-mask constructions for every configuration of at most four emitters, for short-span sources, and for sources with distinct cyclic pair distances. The last class has no span restriction and contains an asymptotically overwhelming fraction of fixed-size supports. For four emitters, ordinary-correlation rigidity and cyclic separation together distinguish reflected configurations. An explicit passive mask also attains the exact optimal Fourier-intensity separation under a prescribed transmission floor. In both the original and complete-reference experiments, sufficiently faint light produces a different law: at fixed sparsity, the optimal photon order grows as the grid size raised to twice the support limit. For at most two emitters, the paper determines the optimal detected-photon order throughout the known-brightness range, from arbitrarily faint light through unit brightness, on odd grids in the original apparatus and on every grid with complete reference collection. The result includes the dependence on grid size, brightness, and error probability, including an additional logarithmic information cost between the faint and bright limits. At fixed confidence, the quadratic regime begins at brightness of order ((\log n)/n^{2}). A mask combining an index ramp with a quadratic-residue code attains the sharp order. On even physical grids, one known vacuum input and odd-dimensional Fourier optics give the same law; omitting one target output preserves the detector count. This is a distinct apparatus. For the specified two-scale transmissions on an infinite even-grid family, an exact lower bound using the complete count record proves that changing phases or decoders alone cannot remove their intermediate-brightness limitation. A collected-light model makes the physical scope explicit. For a known collection operator, binary-source identifiability is equivalent to injectivity of the induced thermal covariance map. Covariance collisions give identical quantum states and therefore prevent discrimination by any subsequent receiver. A passive multiport establishes population identifiability when the map is injective. The paper also derives the exact throughput for uniform passive mode separation from the smallest Gram eigenvalue and accounts separately for detected and incident photons. These results connect the assumed mode access to a specified collection model; they do not establish a general quantum-optimal receiver or a complete astronomical implementation. Reference excess variance supplies a normalization that identifies the spatial source despite an unknown finite common-profile thermal-mode spectrum and independent unknown Poisson backgrounds. The finite-sampling guarantee includes the empty source under explicit mean and nonzero-signal bounds. Population identifiability of the mode spectrum is distinguished from stable estimation. When background means are known, the arm means recover source mass and improve a guaranteed scale-free precision radius. Further results include exact thermal count variances, confidence exponents, calibration bounds, a photon penalty for discarding detector labels, and an exact target-only localization obstruction when singleton brightness is unrestricted and unknown. The results have analytical proofs, supported by 34 reproducible finite and symbolic verification suites. These checks provide corroboration, not experimental validation or external peer review. The unrestricted fixed-sparsity unit-brightness problem is resolved on all grids. Remaining questions include the optimized even-grid intermediate-brightness transition in the original apparatus, transitions for three or more emitters, sharp growing-support rates, sharp reference-coverage dependence at vanishing brightness, and optimal covariance-only scale-free precision. Continuous off-grid recovery requires a tolerance-based criterion and additional stability analysis. The work makes no literature-priority claim.
Quantum illumination uses a retained idler to improve detection of a weak optical return in thermal noise. For a source designer, the practical resource question is how much ideal quantum storage is needed for a specified improvement. This work gives the exact asymptotic answer when the signal-photon budget and the total retained-register dimension are constrained independently. Exact finite-memory frontier Consider a thermal attenuator with transmissivity 0<η<1 and vacuum-output background occupation b>0, tested against the replacer that returns the same thermal state. For n parallel uses, every admissible probe obeys a mean total signal-photon ceiling nN and a strict bound dimR≤⌊enP⌋ on the entire retained register, including retained source records. The memory rate is P/ln2 ideal qubits per use. With vanishing miss probability, the optimal false-alarm exponent is Ξ(N,P)=ηln(1+1/b)N+G(min{P,g(N)}). Here all logarithms are natural, g(x)=(x+1)ln(x+1)−xlnx, with g(0)=0, and G(p)=g(b)+p−F(g−1(p)). The function F(x) is the entropy of the full complementary output for a thermal input of mean occupation x; the complement includes the purification of the thermal environment. The first term in Ξ is the optimum without retained memory, and G is the additional detection benefit. The converse covers arbitrary normal mixed probes, correlations across channel uses and collective measurements. Matching sources satisfy the actual finite-register and energy bounds: they allocate photons between signal–idler correlations and a known displacement, and approach the frontier through finite-support preparations. How much storage is useful? The exact saturation rate and initial marginal value are Psat(N)=g(N),G′(0+)=1+bη. Because G is strictly increasing and strictly concave, at a positive signal budget N>0 the minimum rate retaining a fixed fraction 0<r<1 of the full additive memory benefit is Pr(N)=G−1(rG(g(N)))<rg(N). At low brightness, Pr(N)∼rg(N). At high brightness, this fixed-fraction storage cost remains bounded while the exact-saturation cost diverges. The relative detection benefit simultaneously tends to zero: saving a large fraction of the saturation storage does not imply a large improvement over the total no-memory exponent. Connection to optical communication For the same thermal channel, let C(N,P) denote the classical capacity under the corresponding photon budget and a strict initial maximally entangled Schmidt-rank supply. The two separately defined tasks satisfy Ξ(N,P)−Ξ(N,0)=C(N,P)−C(N,0). This identity equates optimized incremental values, not the physical resources or protocols. The manuscript explicitly attributes the shared Gaussian-reference contraction and entropy-support tools to the capacity analysis. Its detection-specific contribution is the original-register mixed-probe converse, strict finite-register attainment and operational testing limit. Scope. The result concerns parallel, phase-known thermal attenuation, ideal retained storage and the vanishing-miss Stein criterion. It is an asymptotic resource benchmark rather than a finite-sample receiver or a noisy-memory optimum. The integrated article includes the complete proofs and two figure groups showing both storage cost and the value obtained from it.
How many uses of an optical device are needed to certify that it preserves quantum information? This article determines the optimal asymptotic number of coherent-light probes under an average photon budget. Matching upper and lower bounds show that a single optical mode carries an unavoidable logarithmic cost, while every fixed higher mode number follows a different power law. Certification task and resource accounting. The device is an arbitrary normal, memoryless bosonic quantum channel. The task distinguishes the identity from channels at least ε away in the energy-constrained diamond norm, defined using the full trace norm and a signal-energy budget Es. This guarantee includes inputs entangled with an unrestricted reference system. The physical probes are fresh coherent states with classically chosen amplitudes. Adaptive choices, retained quantum outputs and a final joint measurement are allowed. Every device call, including a vacuum pulse, is counted, and the actual illumination obeys ∑t=1TEK∥αt∥2≤TEp under every channel in the testing promise. The probe budget Ep is distinct from the signal budget Es defining the certification guarantee. Optimal mean-energy query laws. For fixed positive Es,Ep, fixed finite mode number m, and both testing errors at most δ, the minimax number of calls satisfies Tmean(1)(ε,δ)=ΘEs,Ep(ε−4logε1logδ1),Tmean(m)(ε,δ)=Θm,Es,Ep(ε−(2m+2)logδ1),m≥2. The bounds hold for sufficiently small ε, uniformly over 0<δ≤1/4. Independent probes drawn from a physical positive multiscale source, followed by individual binary target tests, attain these orders. The matching converse covers adaptive coherent-probe protocols with quantum output memory. Thus those additional processing capabilities do not improve the asymptotic order within the stated preparation model. The result specifies matching scaling laws, rather than exact finite-query counts or optimal numerical constants. Why one mode has an extra logarithm. Rare high-photon coherences can strongly affect an energy-bounded quantum input while leaving only a weak signature on coherent probes. The proof retains this interference through a positive coherent-frame Green kernel. Its spectral sum grows as tlogt for one mode and as tm for fixed m≥2. A complementary family of physical channels shows that the single-mode logarithm is necessary, rather than an artifact of the reconstruction bound. What source brightness changes. The optimal mean-only laws contrast with prescribed independent Gaussian modulation, for which logTG=Θ(ε−2) at fixed confidence. A fixed brightness cap gives logTcap=Θ(ε−2log(1/ε)). The article also establishes matching logarithmic orders under stretched-exponential brightness constraints and develops the conversion from average coherent-probe error to thermal Choi-state and energy-constrained worst-input guarantees. These are worst-case certification limits for the specified coherent-probe access model. The channel class retains infinite photon tails and imposes no output-energy bound. This updated, integrated 37-page article contains the main results, two figure groups, complete proofs and references.
Can a measurement that records only one global vacuum event recover an optical mode spectrum? This article identifies when increasing the number of simultaneously interfering copies supplies exactly the spectral information lost by a two-copy test. It provides a finite algebraic reconstruction without first determining the generating mode basis. Source and measurement. Consider a pure source with known total photon number N≥2 and at most m occupied orthogonal modes: ∣Ψ⟩=∑j=1mwjeiθj∣Nj⟩,wj≥0,∑jwj=1. Here ∣Nj⟩ has all photons in mode j, and the target is the unordered spectrum {wj} of the normalized one-particle reduced state. The source phases and a common passive mode basis may be unknown. This source class includes every pure two-photon state; for N≥3, the generalized NOON/GHZ form is an explicit preparation promise. Interfere ℓ complete, independent, identically prepared copies in a Fourier multiport, acting identically on every physical signal mode. The probability that all relative output modes are vacuum is Mℓ=ℓd∫QΨ(z)ℓdμd(z),QΨ(z)=∣⟨z∣Ψ⟩∣2,dμd=π−dd2dz, where d is the physical mode number. Spectator vacuum modes do not change this probability. Exact reconstruction and its mechanism. The probabilities M2,…,Mm, together with normalization, determine every mode weight, including zero and repeated weights. The new connection is from these optical probabilities to spectral power sums. A formal logarithm separates connected photon pairings; their strictly positive coefficients make the map triangular and invertible at every order. Standard Newton identities then recover the spectrum. For generic positive, distinct weights, the lower prefix through Mm−1 leaves even the largest weight undetermined. A concrete pair of three-photon sources has spectra (1/2,1/2,0) and (2/3,1/6,1/6). Both give M2=11/64, while their three-copy probabilities are 155/6561 and 183/6561. The reconstructed largest weight also determines the coherent peak and full trace-norm distance to a pure coherent state: q=N!NNe−Nmaxjwj,DC2=4(1−q). Physical resource accounting. Highest interference order is distinct from total sample consumption. With nℓ independent trials at order ℓ, the total number of input copies is Cin=∑ℓ=2mℓnℓ, and the incident photon count is NCin. A finite two-hypothesis counting example is included. The exact inverse is not a uniform noisy-spectrum confidence theorem, and the order-necessity result concerns this specified vacuum-event hierarchy rather than arbitrary optical measurements. Mode-resolved one-particle measurements provide a different route to the spectrum. The integrated 13-page article contains two figure groups, complete proofs and references. It distinguishes the all-order optical-to-spectral connection from established replica-moment formulas and classical power-sum inversion.
What can be transmitted with a fixed photon supply? An optical source that supplies exactly K photons across n modes poses a concrete communication problem: how many messages can it transmit at a chosen error probability, and which input states make the best use of those photons? This paper gives directly evaluable coding bounds for pure loss and identifies the exact optimal input ensembles for identical noisy Gaussian channels. The results allow researchers to assess finite optical blocks, quantify the information cost of fixing the photon supply, and determine when entanglement between separately constrained blocks can be dispensed with. Every message is supported on the same exact K-photon sector. Within that sector, encoding may use arbitrary mixed states and entanglement among modes, and the receiver may perform a joint measurement. Communication is unassisted. All logarithms below are natural; information is measured in nats. Finite-message bounds and the optimal third-order rate For n ≥ 2 identical pure-loss channels of transmissivity 0 < η < 1 and K ≥ 1, the number L of surviving photons has the same binomial distribution for every encoded state. Define pℓ=(ℓK)ηℓ(1−η)K−ℓ,dn,ℓ=(ℓn+ℓ−1). If ps*(M;n,K) is the best average success probability for M equiprobable messages, then ∑ℓ=0Kpℓ[1−dn,ℓM−1]+≤ps∗(M;n,K)≤∑ℓ=0Kpℓmin{1,Mdn,ℓ}. Here [x]+ = max{x,0}. The lower bound provides an achievable guarantee; the upper bound limits every admissible code. Evaluating both sums brackets the performance at a specified photon supply and block length. Their proof has a simple physical interpretation: surviving photons determine the available output dimension, while one collective-mode codebook works across all possible survivor counts. For fixed E > 0, fixed error tolerance 0 < ε < 1, Kn = ⌊nE⌋ and en = Kn/n, the bounds determine the largest message count through third order: logM∗(n,Kn,ε)=ng(ηen)+nη(1−η)enh(ηen)ΦN−1(ε)−21logn+O(1). Here g(x) = (x+1)log(x+1) − x log x, h(x) = log(1+1/x), and ΦN is the standard normal distribution function. This fixes both the fluctuation scale and the optimal −½ log n correction. The finite bounds also yield the exact decay exponent of success above capacity and a matching error exponent on a specified interval below capacity. Which inputs are optimal under Gaussian noise? For n identical nonidentity gauge-covariant Gaussian channels, with thermal photon means transformed as r ↦ ar+b, a > 0 and b ≥ max{0,a−1}, the paper determines the exact fixed-photon Holevo optimum and every ensemble attaining it. Each optimal letter places all K photons in one collective mode: ∣K,u⟩=K!(∑j=1nujaj†)K∣0⟩,∑j∣uj∣2=1. An ensemble is optimal precisely when almost all its letters have this form and its average is uniform on the K-photon sector. A finite weighted optimal ensemble exists. These collective modes may themselves extend across, and be entangled among, physical optical modes. Thus the characterization identifies the structure to optimize without imposing a Gaussian-input restriction. The central proof step combines different loss histories that end in the same output photon sector. A common ordering of the comparison eigenstates allows the separate majorization bounds to be combined. For pure loss, a quantitative rigidity estimate further converts a small Holevo deficit into closeness to collective-mode number states. Resource cost and communication across blocks At fixed positive photon density and fixed nonidentity Gaussian noise, requiring an exact photon supply incurs a Holevo penalty of ½ log n plus an explicitly determined constant, relative to the usual mean-energy value. The paper separates this cost into photon-count uncertainty and the mismatch from a thermal output distribution. For separately constrained blocks of quantum-limited loss, quantum-limited amplification or entanglement-breaking Gaussian noise, the one-block Holevo values add exactly. Every optimal letter is a product of blockwise collective-mode number states, and the repeated-block unassisted classical capacity equals the one-block Holevo value. The low-rate reliability interval outside the matching range and repeated-block additivity for intermediate Gaussian noise remain open.
What should an optical memory retain when only a limited amount of source information can be kept? This paper gives the complete quantum information frontier for a fixed Gaussian source and relevance channel, and identifies when a single retained qubit preserves correlations that every classical record loses. It also turns that advantage into a concrete test of whether received light belongs to its stored source preparation. The physical experiment and the complete frontier A fixed Gaussian state on A is prepared with its actual purifier C. A normal encoder acts on C to produce memory M, while a prescribed Gaussian channel maps A to the relevant optical output B. The source-information cost and useful correlation are r=I(A:M) and v=I(B:M), respectively. All logarithms are natural. The paper determines CQ(R)=supN:I(A:M)≤RI(B:M). For every fixed faithful finite-mode Gaussian source and finite-mode Gaussian relevance channel, optimization over arbitrary normal quantum encoders reduces to Gaussian supporting values and a mixture of at most two Gaussian branches. The branch flag is physically retained. The unrestricted frontier is a supremum with no imposed memory dimension or memory-energy cap; an explicit finite approximation gives an error uniform in the information budget. Hard finite-dimensional implementations are treated separately. Why one qubit can preserve the leading low-brightness information For a thermal source with mean photon number n and a thermal attenuator with transmissivity 0<η<1, write the output occupation as ηn+b, where b≥0 is the environment's added thermal occupation. With g(x)=(x+1)log(x+1)−xlogx, the exact optimum over all normal classical records is Ccl(n,R)=g(ηn+b)−g(ηuR+b),uR=g−1(max{g(n)−R,0}). A legal encoder applies vacuum loss with transmissivity λ to the purifier and then coherently retains two levels, resetting the entire higher-photon tail. For fixed b>0 and 0<λ<1/2, at the common budget Rn=g(n)+g(λn)−g((1−λ)n), its actual relevance satisfies v2=1+bηλnlog(1/n)+O(n),Ccl(n,Rn)=2ληlog(1+1/b)n+O(log(1/n)n). The full optical tail is included in the resource accounting. At n=0.01, η=b=0.5, λ=0.2 and R=0.023875, the certified predictive-mutual-information advantage exceeds 85%. The leading correlation is quantified by memory-side discord and can survive when the final output–memory state is separable. The vacuum-environment endpoint is evaluated separately for the same encoder. A same-resource correlation decision The receiver distinguishes matched source pairs from independent systems with identical marginals. For k uses, the retained predictive memory obeys dimKk≤2k and I(Ak:Kk)≤kR, including all retained flags. Both competitors may make arbitrary joint binary measurements on the received optical modes and their retained memory. The received modes are common observation access, outside the predictive-memory cap. Let αk be the missed-correlation probability and βk the false-match probability. For every classical block encoder, even with an unrestricted classical alphabet, the fixed-error converse is αk≤ϵ⟹βk≥exp[−1−ϵkcnR+log2],cn=ηlog(1+1/n)log(1+1/(ηn+b)). At the finite parameters above and ϵ=0.1, the product-qubit encoder achieves an asymptotic false-match exponent of 0.0049 nats per pair, whereas every classical block strategy has exponent below 0.003138386231. This uses one qubit per source pair and identical measurement permissions. It is an asymptotic discrimination guarantee, distinct from the 85% single-copy information comparison; it specifies neither a finite-sample threshold nor a laboratory receiver circuit. The integrated paper contains the complete proofs, resource accounting and figures. The exact thermal classical comparator, the finite-memory construction and the block-testing connection have explicit proof dependencies separate from the general Gaussian-frontier reduction.
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)≤cIfor 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 −lnc. 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)=1−ηd−cd(d+1)/2.NG(c,η)=1−ηc−1−1. 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∗=(d entriesc,…,c,r). It majorizes every output of the same channel, so the minimum von Neumann entropy is exactly Smin(Φη,σ)=−dclnc−rlnr. 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≥mlnj 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=∣1⟩⟨1∣, 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 2m, 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.