How many uses of an optical channel are required to detect weak phase noise when only the actual mean number of submitted photons is bounded? This paper determines the optimal fixed-confidence order, including every causal protocol with quantum memory and feedback. The two hypotheses are ideal transmission and independent Gaussian phase diffusion of known variance \(\gamma>0\). In the photon-number basis the noisy channel is \[\mathcal G_\gamma(|n\rangle\langle\ell|)=e^{-\gamma(n-\ell)^2/2}|n\rangle\langle\ell|.\] A protocol has at most \(T\) channel calls. Under each hypothesis separately, the sum of unconditional mean photons submitted over all calls is at most \(TE\). Reused photons are charged on every submission, including failed or discarded branches. Inputs may be arbitrary normal states with infinite photon tails, entangled across calls and with retained reference systems. Let \(T_\delta^{\mathrm{causal}}(E,\gamma)\) be the minimum call count for both decision errors to be at most a fixed \(0<\delta<1/2\). For \(0<\gamma\le1\) and \(0<E\sqrt\gamma\le1/4\), \[T_\delta^{\mathrm{causal}}(E,\gamma)=\Theta_\delta\!\left((E\sqrt\gamma)^{-1}\right).\] The explicit bounds are \[\frac{\sqrt2[1-2\sqrt{\delta(1-\delta)}]}{E\sqrt\gamma}\le T_\delta^{\mathrm{causal}}(E,\gamma)\le\left\lceil\frac{\log(1/\delta)}{b_0E\sqrt\gamma}\right\rceil,\qquad b_0=\tfrac34(1-e^{-1/2}).\] A common-environment overlap bound charges each loss of fidelity to the two actual hypothesis-dependent photon budgets, giving the converse even with causal feedback. Independent vacuum–Fock superpositions with \(n=\lceil\gamma^{-1/2}\rceil\), photon probability \(p=E/n\), and immediate binary measurements attain the order. The powers match at fixed confidence; the numerical constants and vanishing-error dependence are distinct. The paper also separates information measures under the same mean-only constraint. The maximum single-use Umegaki relative-entropy disturbance, allowing any reference on a separable Hilbert space, is of order \(s\log(1/s)\) for \(s=E\sqrt\gamma\downarrow0\) in the stated regime. The corresponding energy-constrained Belavkin–Staszewski divergence is infinite. The entropy statement concerns single-use disturbance, while the detection theorem covers complete causal strategies. Vacuum–Fock probes and their actual photon accounting are inherited from established dephasing-discrimination work; the matching causal converse and reusable overlap and fractional-moment controls are developed with their full resource conditions. Retained memory and controller complexity are unrestricted. The integrated 11-page PDF contains two figures, complete proofs and eight references.
How many uses of an optical channel are required to detect weak phase noise when only the actual mean number of submitted photons is bounded? This paper determines the optimal fixed-confidence order, including every causal protocol with quantum memory and feedback. The two hypotheses are ideal transmission and independent Gaussian phase diffusion of known variance (\gamma>0). In the photon-number basis the noisy channel is [\mathcal G_\gamma(|n\rangle\langle\ell|)=e^{-\gamma(n-\ell)^2/2}|n\rangle\langle\ell|.] A protocol has at most (T) channel calls. Under each hypothesis separately, the sum of unconditional mean photons submitted over all calls is at most (TE). Reused photons are charged on every submission, including failed or discarded branches. Inputs may be arbitrary normal states with infinite photon tails, entangled across calls and with retained reference systems. Let (T_\delta^{\mathrm{causal}}(E,\gamma)) be the minimum call count for both decision errors to be at most a fixed (0<\delta<1/2). For (0<\gamma\le1) and (0<E\sqrt\gamma\le1/4), [T_\delta^{\mathrm{causal}}(E,\gamma)=\Theta_\delta!\left((E\sqrt\gamma)^{-1}\right).] The explicit bounds are [\frac{\sqrt2[1-2\sqrt{\delta(1-\delta)}]}{E\sqrt\gamma}\le T_\delta^{\mathrm{causal}}(E,\gamma)\le\left\lceil\frac{\log(1/\delta)}{b_0E\sqrt\gamma}\right\rceil,\qquad b_0=\tfrac34(1-e^{-1/2}).] A common-environment overlap bound charges each loss of fidelity to the two actual hypothesis-dependent photon budgets, giving the converse even with causal feedback. Independent vacuum–Fock superpositions with (n=\lceil\gamma^{-1/2}\rceil), photon probability (p=E/n), and immediate binary measurements attain the order. The powers match at fixed confidence; the numerical constants and vanishing-error dependence are distinct. The paper also separates information measures under the same mean-only constraint. The maximum single-use Umegaki relative-entropy disturbance, allowing any reference on a separable Hilbert space, is of order (s\log(1/s)) for (s=E\sqrt\gamma\downarrow0) in the stated regime. The corresponding energy-constrained Belavkin–Staszewski divergence is infinite. The entropy statement concerns single-use disturbance, while the detection theorem covers complete causal strategies. Vacuum–Fock probes and their actual photon accounting are inherited from established dephasing-discrimination work; the matching causal converse and reusable overlap and fractional-moment controls are developed with their full resource conditions. Retained memory and controller complexity are unrestricted. The integrated 11-page PDF contains two figures, complete proofs and eight references.
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.
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.
How does the shape of detector noise affect optical communication? Two receivers can have the same noise covariance yet support different communication rates. This work determines both the quantum noise floor and the photon scale for approaching the capacity ceiling of specified non-Gaussian optical measurements. The central distinction is between the size of a high-energy communication advantage and the finite signal energy needed to realize it. Exact quantum noise floor. The receiver is a displacement-covariant measurement with a fixed, unflagged ancillary state. For every vacuum–single-photon seed σt=(1−t)∣0⟩⟨0∣+t∣1⟩⟨1∣, with 0≤t≤1/2, the output Shannon entropy is minimized uniquely by coherent states over all normal quantum inputs, without an input energy or photon-number cutoff. A separate entropy theorem covers three-level seeds σ=p0∣0⟩⟨0∣+p1∣1⟩⟨1∣+p2∣2⟩⟨2∣ with p1≤0.1, p2≤0.001, and p0=1−p1−p2. A capacity decomposition with a matched thermal benchmark. Let E be the mean signal photon budget per use, μ the fixed mean ancillary photon number, k the coherent-input response, and G1+μ the circular Gaussian density with the same second moment. With natural logarithms, the unassisted classical capacity satisfies Cσ(E)=ln(1+1+μE)+D(k∥G1+μ)−Δσ(E). The first term is the exact capacity of the matched thermal receiver, allowing arbitrary quantum encoders. The relative entropy is the limiting high-energy capacity gain. The deficit Δσ(E)=Uσ(E)−Cσ(E), where Uσ(E)=1+ln(E+1+μ)−h(k), is the finite-energy shortfall from that gain. The converse permits mixed, non-Gaussian and entangled block codewords under an average total signal-photon constraint. Response zeros determine the sharp convergence rate. For each fixed nonvacuum seed in the stated families, write the coherent-response transform as k(ξ)=e−uP(u), where u=∣ξ∣2, P(u)=1−μu+vu2, and v=p2/2. Then limE→∞E−lnΔσ(E)=2minP(ζ)=0∣ζ∣. Positive classical modulation of coherent states attains this exponent with exact signal-energy accounting. In the nonvacuum two-level family, ordinary Gaussian modulation instead leaves an algebraic deficit. Uniform estimates cover compact three-level parameter families through the merger of two real zeros into a complex pair. Thus the response's relative entropy determines the high-energy gain, while its zeros determine the leading photon cost of approaching that gain. Finite-energy guarantees. Explicit bounds compare ordinary Gaussian modulation with a positive variance-mixture preparation and an upper bound valid for every quantum code. For t=1/2 and E=1, the capacity gain over the thermal receiver with the same ancillary mean photon number obeys 0.009742<Cσ1/2(1)−Cτ1/2(1)<0.010469nat/use. The better achievable bound depends on signal energy; no exact finite-energy optimizer is assumed. The matched-covariance noise comparison builds on classical additive-noise results of Ihara and Diggavi–Cover. The contribution here connects an exact all-input quantum noise floor to a sharp capacity remainder and positive, energy-accounted preparations. The results concern the stated receiver families and ideal measurement model; ancillary preparation and implementation overhead are separate resources. The integrated article includes complete proofs and five figures.
How much quantum information must a sequential optical test retain, and how reliably must that information survive between source preparations? This paper gives preparation budgets and impossibility bounds that distinguish useful quantum storage from regimes where the classical sample order is unavoidable. Optical source and fully counted memory A fixed unknown normal state supplies independent, identical preparations on m bosonic modes. The null consists of all pure Gaussian states with total mean photon number at most one. Alternatives have half trace distance at least 1/32 from every pure Gaussian state; neither their energy nor the comparison family's energy is capped. Mixed states and infinite photon tails are included. Each visit accesses one entire original once. The total quantum dimension retained between visits is at most D, while classical records and computation are unrestricted. At every actual waiting interval, the entire register independently survives unchanged with known probability a, or is erased with a classical flag. The quantity ka(D,m) is the smallest deterministic original-preparation cap achieving worst-case error at most 1/3. Failed preparations, discarded trials and reserved positions are all charged. A retained register of q qubits has D=2q. An exact threshold for statistical advantage Without retained quantum information, the optimal sample order is Θ(m). Along arbitrary growing-mode sequences, the precise criterion for improving that order is ka(D,m)=o(m)⟺am→∞andmaD→∞. The two conditions express separate capacity and survival requirements. Arbitrarily large storage cannot reduce the linear sample order when a=O(1/m), yet a survival probability tending to zero can still permit an advantage. Let A=max{1,m2/D}. At the fixed energy and separation above, constants c,C>0, independent of m,D,a, give cmin{m,A/a}≤ka(D,m)≤Cmin{m,A/a+min(a−2,m2/3)},0<a≤1. At a=0, the sample cost is Θ(m). For ideal memory the complete rate is k1(D,m)=Θ(min{m,max(1,m2/D)}). With loss, the bounds match whenever A≥min{a−1,am2/3}. For example, a=m−1/6 and D=⌈m11/6⌉ give Θ(m1/3) preparations. The existence of sublinear advantage is fully characterized even where its optimal exponent remains bracketed: for D≥m2, a=m−1/2, the stated bounds are Ω(m1/2) and O(m2/3). Why the construction and converse fit the same resources A physical completely positive contraction combines first- and second-moment information before the next wait. Although the temporary pair output has order m2 labels, only a positive register of dimension m+1 is retained. The three-visit route requires D≥m+1 and pays two actual survival events; a classical alternative handles the centering statistics. The converse covers arbitrary adaptive instruments and persistent noncommuting memory. Erasure removes the coherent prediction contribution while preserving information already learned in the full classical history. Unflagged noise and applicability For a fixed memoryless channel on the entire retained register with a physical reset decomposition N=aΨ+(1−a)Rτ, where Ψ is CPTP and Rτ(X)=Tr(X)τ, an exact flagged-erasure simulation gives kN(D,m)≥ka(D,m). The established positive quantum Doeblin coefficient optimizes this lower obstruction. Whole-register depolarizing noise with unchanged fraction 1−q=O(1/m) therefore forces Θ(m) samples for every memory dimension. General unflagged noise inherits a converse, not the flagged model's complete achievability guarantees. The results quantify asymptotic preparation and persistent-memory resources under ideal visit-local control. Temporary auxiliary systems and gate complexity have no imposed budget, and the explicit sufficient finite-sample constants are conservative. The task is pure Gaussianity testing, rather than membership in the full family of mixed Gaussian states. The single integrated PDF contains the short main text, full proofs, four figures and precise resource accounting.
How much quantum memory is needed to retain the contributions of rare photon events to a measurement record? This paper determines the memory scales required to approach the best heterodyne conditional entropy when the total optical energy tends to zero. Its resource laws distinguish precisions for which counting photon configurations suffices from critical precisions at which energy-calibrated entropy corrections determine additional memory costs. Full preparation model and reference value. The optical system has m bosonic modes, total mean photon number at most E, and an entire accessible quantum side system B of dimension at most D, including retained preparation labels. The optical source and its correlations with the memory are optimized over all normal mixed states; arbitrary coherences and infinite photon tails are allowed. With natural logarithms and heterodyne measure dμ=π−md2mz, define Vm,D(E)=minρABHμ(Z∣B),Hμ(Z∣B)=hμ(Z)−χ(Z:B). Here Z is the heterodyne outcome and χ is Holevo information. The unlimited-memory optimum at the same energy is Bm(E)=m−Eln(1+m/E),Dmin(m,E,ε)=min{D:Vm,D(E)−Bm(E)≤ε}. The tolerance ε is a total conditional-entropy excess in nats. It is not a measurement-error probability. Classical side information satisfies Hμ(Z∣Y)≥m; entangled quantum side information can go below this benchmark. The resource D is a Hilbert-space dimension, corresponding to ⌈log2D⌉ qubits. Precision selects a photon-number hierarchy. Along arbitrary joint limits m→∞, E→0, with no prescribed relation between E and 1/m, Eε→c>0⟹Dmin∼e−cm. For every fixed integer K≥2, EKε→c∈(0,1/K!)⟹Dmin∼(1−K!c)K!mK. Although higher-photon events become rarer, they occupy increasingly many orthogonal mode configurations. Resolving their entropy contribution therefore requires a larger memory dimension. In qubits, the fixed-K law is Klog2m+O(1). Critical precisions resolve costs beyond sector counting. At the exact tolerance E2/2, the complete vacuum–one-photon head is not sufficient at leading dimension: Dmin(m,E,E2/2)=m+1+(1+o(1))(2m+6Em2). The error is relative to the additional dimension in parentheses. Both terms come from the entropy of the mean-calibrated finite head, together with an all-state output-entropy comparison. At the next critical tolerance E3/6, the paper determines an added-to-removed-level transition near the complete two-photon head (2m+2), including the boundary near mE=4 and precisely qualified integer conclusions. An exact finite-energy source. For every finite m, 2≤D≤m+1, and 0<E≤10−4, let r=D−1 and choose orthonormal one-photon modes u1,…,ur. The preparation ∣Ψ⟩=1−E∣0⟩A∣0⟩B+E/r∑j=1r∣uj⟩A∣j⟩B attains the original global minimum over the full preparation class. Its vacuum–one-photon support is a conclusion of the optimization, rather than an imposed photon cutoff. The paper also characterizes exact small-energy optima for every fixed complete photon head. A record-reconstruction interpretation with explicit resource accounting. A common finite heterodyne digitizer, including a genuine overflow symbol, approximates the conditional entropy uniformly over all legal preparations. For independent repetitions of a known preparation, classical–quantum Slepian–Wolf coding identifies the asymptotic reconstruction rate with H(X∣B). The uniform comparison transfers the hierarchy to the excess optimized communication rate relative to unlimited finite memory using the same digitizer. Preserving the first critical formula requires digitization error ξ=o(E2/m+E3); the corresponding error scales for the separated second-critical regimes are also given. The decoder may act collectively after receiving the classical message. For n independent preparations, the supplied side-system dimension is at most Dn and total mean optical energy is at most nE. This is a supplied-memory resource law, not an optimization over arbitrary correlated block preparations, a minimum Schumacher-compressed storage rate, or a bound on peak decoder workspace. Coding limits are taken at fixed finite parameters before the low-energy, growing-mode limit. The work develops the low-photon, growing-mode regime of the preparation problem studied in Optimal entanglement weights for heterodyne uncertainty with finite quantum memory (doi:10.5281/zenodo.23647071). The integrated 29-page paper contains the complete proofs, three figures and two tables.
How much quantum memory is needed to restore an unknown optical pulse together with its correlations? This paper gives matching storage laws showing that the answer depends on the actual joint preparation, even when the optical marginal is a positive mixture of coherent states. Fix an optical mode number m, a known real k-dimensional displacement subspace with 1≤k≤2m, and a positive mean photon budget U0. The source is any probability mixture of coherent states in that subspace with actual mean photons at most U0. One normal encoder and decoder must work for the entire unknown source class, and every transmitted message or flag counts toward the same register dimension. The reference is inaccessible to both maps. Let δ be the required half trace-distance error and qmin(δ) the least encoded qubit count. As δ↓0, qmin(δ)=⎩⎨⎧2klog2(1/δ)+O(1),klog2(1/δ)+O(1),optical marginal or joint coherent preparation,arbitrary reference extension or classical-bit reference. A single inaccessible classical bit already witnesses the larger leading cost. To recover the smaller cost for correlations, the joint source must actually be prepared as ωAB=∫∣Jx⟩⟨Jx∣⊗τB(x)dμ(x), with ∫∣x∣2dμ≤U0. Here J is the known displacement embedding and τB(x) may be noncommuting normal reference states. The constants are uniform in the reference dimension and the orientation of J; the codec may depend on the known J. The mechanism is the difference between the probability of a rare bright component and the square-root amplitude of its retained correlations. Exact optical Gram matrices and a reference-coherence witness establish the converse for arbitrary normal decoders, including infinite-rank outputs. A common finite optical subspace supplies the attaining compression schemes while retaining the full photon tails. These are single-input universal memory-dimension laws at fixed m,k,U0, with every communication resource counted. They complement established known-source and many-copy compression theory. The integrated 10-page PDF contains two figures, complete proofs and references.