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 \(\varepsilon\) away in the energy-constrained diamond norm, defined using the full trace norm and a signal-energy budget \(E_s\). 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 \[\sum_{t=1}^{T}\mathbb E_{\mathcal K}\|\alpha_t\|^2\le T E_p\] under every channel in the testing promise. The probe budget \(E_p\) is distinct from the signal budget \(E_s\) defining the certification guarantee. Optimal mean-energy query laws. For fixed positive \(E_s,E_p\), fixed finite mode number \(m\), and both testing errors at most \(\delta\), the minimax number of calls satisfies \[T_{\mathrm{mean}}^{(1)}(\varepsilon,\delta)=\Theta_{E_s,E_p}\!\left(\varepsilon^{-4}\log\frac1\varepsilon\log\frac1\delta\right),\] \[T_{\mathrm{mean}}^{(m)}(\varepsilon,\delta)=\Theta_{m,E_s,E_p}\!\left(\varepsilon^{-(2m+2)}\log\frac1\delta\right),\qquad m\ge2.\] The bounds hold for sufficiently small \(\varepsilon\), uniformly over \(0<\delta\le1/4\). 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 \(t\log t\) for one mode and as \(t^m\) for fixed \(m\ge2\). 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 \(\log T_{\mathrm G}=\Theta(\varepsilon^{-2})\) at fixed confidence. A fixed brightness cap gives \(\log T_{\mathrm{cap}}=\Theta(\varepsilon^{-2}\log(1/\varepsilon))\). 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.
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 (\varepsilon) away in the energy-constrained diamond norm, defined using the full trace norm and a signal-energy budget (E_s). 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 [\sum_{t=1}^{T}\mathbb E_{\mathcal K}|\alpha_t|^2\le T E_p] under every channel in the testing promise. The probe budget (E_p) is distinct from the signal budget (E_s) defining the certification guarantee. Optimal mean-energy query laws. For fixed positive (E_s,E_p), fixed finite mode number (m), and both testing errors at most (\delta), the minimax number of calls satisfies [T_{\mathrm{mean}}^{(1)}(\varepsilon,\delta)=\Theta_{E_s,E_p}!\left(\varepsilon^{-4}\log\frac1\varepsilon\log\frac1\delta\right),] [T_{\mathrm{mean}}^{(m)}(\varepsilon,\delta)=\Theta_{m,E_s,E_p}!\left(\varepsilon^{-(2m+2)}\log\frac1\delta\right),\qquad m\ge2.] The bounds hold for sufficiently small (\varepsilon), uniformly over (0<\delta\le1/4). 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 (t\log t) for one mode and as (t^m) for fixed (m\ge2). 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 (\log T_{\mathrm G}=\Theta(\varepsilon^{-2})) at fixed confidence. A fixed brightness cap gives (\log T_{\mathrm{cap}}=\Theta(\varepsilon^{-2}\log(1/\varepsilon))). 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.
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.
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.
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.
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 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 γ>0. In the photon-number basis the noisy channel is Gγ(∣n⟩⟨ℓ∣)=e−γ(n−ℓ)2/2∣n⟩⟨ℓ∣. 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 Tδcausal(E,γ) be the minimum call count for both decision errors to be at most a fixed 0<δ<1/2. For 0<γ≤1 and 0<Eγ≤1/4, Tδcausal(E,γ)=Θδ((Eγ)−1). The explicit bounds are Eγ2[1−2δ(1−δ)]≤Tδcausal(E,γ)≤⌈b0Eγlog(1/δ)⌉,b0=43(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=⌈γ−1/2⌉, 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 slog(1/s) for s=Eγ↓0 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 much high-photon quantum coherence is controlled by a nonnegative phase-space distribution? This article gives quantitative guarantees for approximating an optical state by a finite photon-number block, and constructs physical states showing why the guarantee cannot be inferred from mean energy or Wigner positivity alone at the strength available for a Gaussian-mixture source. Source classes and scope. The comparison concerns one optical mode with [Q,P]=i, mean photon number at most E>0, and a known parity centre. The full class WE consists of normal Wigner-positive states at the exact vacuum min-entropy endpoint, equivalently states supported on the even photon sector in that frame. The subclass GE consists of actual probability mixtures of centred pure squeezed vacua. Mixture labels are unavailable, component energies may be arbitrarily large, and their true average energy is charged. Infinite photon tails are retained. A hierarchy of coherent-tail guarantees. Let PL project onto photon numbers at most L, and define the cutoff error by eL(ρ)=21∥ρ−PLρPL∥1. This quantity includes both discarded photon population and coherence across the cutoff. A mean-energy constraint alone gives the standard OE(L−1/2) guarantee. At the exact Wigner endpoint, the article proves supρ∈WEeL(ρ)≤CE(L+1)−3/4,supρ∈GEeL(ρ)≤L+14E,L≥2. The stronger endpoint bound combines operator positivity and Wigner positivity on the same density kernel, followed by a controlled conversion to trace norm. It requires only mean energy, without a uniform second photon moment. Physical coherence beyond the Gaussian-mixture bound. A family τd∈WE has exact energy E, covariance (E+1/2)I, strictly positive Wigner functions and nonzero population at every even photon number. For cutoffs Ld=Θ(d3), eLd(τd)≥cEd−5/2,(Ld+1)eLd(τd)⟶∞. Thus the Gaussian-mixture L−1 law fails uniformly on the full endpoint class. The construction preserves coherence through an actual optical partial trace and pays its physical energy cost. Positive phase mixing further yields distinct states with identical complete photon statistics and covariance but nearly indistinguishable bounded optical responses. The rare bright branches obey the prescribed mean-energy budget; no efficient experimental preparation protocol or uniform branch-energy bound is assumed. Consequences for a specified measurement system. Each independent copy is measured by bounded centred squeezing followed by a vacuum/nonvacuum binary test. At fixed energy, squeezing range and confidence, let nX(δ) be the minimum worst-case number of copies for trace-distance accuracy δ over source class X. As δ↓0, the bounds are lognGE(δ)=Θ(δ−1),cδ−6/5≤lognWE(δ)≤Cδ−4/3. These are measurement-restricted reconstruction costs. The lower bounds allow adaptive randomized settings. The upper bounds use finite designs and estimates in the original source class, with the energy budget respected for every record; they establish statistical existence rather than an efficient catalogue-generation algorithm. The full-class exponents remain distinct lower and upper bounds. The article also characterizes infinite-order Wigner entropy equality by displaced even-parity support and corrects the pure-Gaussian-only equality clause at infinite order in Dias and Prata (2023). The 27-page integrated article includes the complete proofs, figures and references.