How long quantum information survives can matter more than how many samples a receiver processes at once. We establish this distinction for detecting optical squeezing with an unknown source and noisy direction records. At fixed five-percent errors, mean input photon number at most 0.006 and transmissivity in [0.1, 0.2], a receiver with a fixed-width persistent register attains weak-signal sample coefficient 7.35. Arbitrary adaptive receivers that erase all quantum information between blocks sublinear in the total sample budget require at least 7.7216; one-copy measurements with unlimited classical feedback require at least 10.3. The separation persists for a disclosed Gaussian thermal source: the reset-block lower bound is 7.5486, still above the persistent construction. Source uncertainty is governed by a separate exact energy boundary. With noisy direction records, coherence and excess photon population together determine a matched sample-order law. The constructive receiver has 4096 active memory levels, a deterministic non-Gaussian interface, and explicit trial and cumulative memory-noise budgets. Thus growing temporary measurement blocks cannot replace persistent quantum storage at the demonstrated detection cost. The results specify when control records are essential and when receiver design must preserve quantum information across a macroscopic fraction of the acquisition.
How long quantum information survives can matter more than how many samples a receiver processes at once. We establish this distinction for detecting optical squeezing with an unknown source and noisy direction records. At fixed five-percent errors, mean input photon number at most 0.006 and transmissivity in [0.1, 0.2], a receiver with a fixed-width persistent register attains weak-signal sample coefficient 7.35. Arbitrary adaptive receivers that erase all quantum information between blocks sublinear in the total sample budget require at least 7.7216; one-copy measurements with unlimited classical feedback require at least 10.3. The separation persists for a disclosed Gaussian thermal source: the reset-block lower bound is 7.5486, still above the persistent construction. Source uncertainty is governed by a separate exact energy boundary. With noisy direction records, coherence and excess photon population together determine a matched sample-order law. The constructive receiver has 4096 active memory levels, a deterministic non-Gaussian interface, and explicit trial and cumulative memory-noise budgets. Thus growing temporary measurement blocks cannot replace persistent quantum storage at the demonstrated detection cost. The results specify when control records are essential and when receiver design must preserve quantum information across a macroscopic fraction of the acquisition.
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.
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.
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.
Can a larger quantum memory compensate for a less reliable one? This paper answers that question for optical heterodyne quantum conditional entropy under a total photon budget and a strict physical memory dimension. It separates an uncertainty loss that additional storage can reduce from an erasure floor that no increase in dimension can remove. Preparation and noise model. The optical system has m modes and total mean photon number at most E. Its joint preparation with a memory is optimized over all normal mixed states, including arbitrary optical coherence and infinite photon tails. The input memory has dimension at most d. Independently, the entire register survives with probability 1−p or is erased into an orthogonal readable failure state. The physical output therefore has d+1 levels; a total q-qubit register permits d≤2q−1. Every event is retained, with no postselection. For heterodyne measure d2mz/πm and natural logarithms, the exact all-preparation reduction is Ver(m,d,E,p)=minrankρ≤d,Tr(Nρ)≤E{W(ρ)−(1−p)S(ρ)}. Here W is Wehrl entropy, S is von Neumann entropy and N is total photon number. The rank constraint follows from pure attainment of the original mixed optical–memory problem. The objective describes quantum side information; it is not automatically the uncertainty achieved by a classical readout of the memory. An exact reliability ceiling. Define g(x)=(1+x)ln(1+x)−xlnx, S∗=mg(E/m), the ideal unlimited-memory value Bm(E)=m−Eln(1+m/E), and aE=1−[(1+E/m)ln(1+m/E)]−1. For any target excess satisfying 0<ε<aES∗, pcrit=mg(E/m)ε. A finite register reaches Ver≤Bm(E)+ε if and only if p<pcrit. At equality only an infinite memory attains the target. Above the ceiling the target is impossible even with unlimited memory. Finite storage and reliability pay different costs. Let Sd be the largest entropy of any rank-at-most-d optical state within the same photon budget, and Δd=S∗−Sd. For 0≤p<aE, pS∗+(aE−p)Δd≤Ver−Bm(E)≤pS∗+(1−p)Δd. The first contribution survives at unlimited dimension, whereas Δd tends to zero as storage grows. An exact finite Gibbs program computes Sd, including partially occupied photon shells. The two bounds give necessary and sufficient resource tests rather than an exact full finite-dimensional frontier. The unlimited-memory floor is obtained from an explicit scalar thermal optimization. Above its energy-use crossover the attaining thermal source uses less than the allowed energy. At 0<E≤10−4, 2≤d≤m+1 and p<1/8, an explicit vacuum–one-photon entangled preparation remains globally optimal over the full preparation class. Finite examples also reserve the failure flag inside the stated register and distinguish heralded erasure from unheralded replacement. The analysis builds on Gaussian Wehrl extremality theorems established in earlier literature and on Zixuan He's previous results on ideal-memory spectrum shaping and low-photon resource laws. Its additional contribution is the optimized finite-memory erasure contract, the exact feasibility boundary and physically budgeted attaining sources. The integrated 14-page article contains the complete proofs, two figure groups, finite certificates and references.
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.
We study the persistent quantum and classical memory required for exact heralded storage and retrieval of quantum measurements. Given only black-box access to the measurement channel, we construct an exact compression of the port-based program into an outcome-type record and a quantum register, with total dimension polynomial in the query count. For every universal retrieval processor we prove the bounds Q+C≥log2n and 2Q+C≥log2((n−1)d2+1) on the numbers Q of qubits and C of bits of the program, for n-outcome POVMs on Cd with d≥2. For n≥2, no finite classical memory removes the need for quantum memory: any such processor requires a quantum program of dimension at least d, at that dimension its classical alphabet must have at least n symbols, and both minima are attained by a single-query protocol with success 1/d2. Combining direct-program compression with a converse from approximate measurement programming gives logarithmic total-memory scaling in 1/(1−p) as the uniform success probability p approaches one, at fixed d,n≥2. At fixed input dimension and target success below one, the optimal total-memory scaling is logarithmic in n. We also construct self-contained programs under branch-decomposition access and compare the quantum and classical costs of all constructions, including where the outcome-type record is kept: for POVMs with n nonzero rank-one effects the direct program admits a purely quantum allocation whose register is never larger than that of the minimal-Naimark unitary program and which succeeds more often whenever n>d.