Can a finite-rank quantum state approximate a given optical state while reproducing an entire continuous measurement record and its original mean energy exactly? This paper provides a constructive affirmative answer, with quantitative rank, trace-distance and entropy control. The main result concerns one oscillator \(A=L^2(\mathbb R)\), with \([Q,P]=i\) and photon operator \(N=(Q^2+P^2-1)/2\), coupled to a finite memory \(B=\mathbb C^b\). The input is any normal state \(\rho_{AB}\) with mean photons \(E=\operatorname{Tr}(N\rho_A)<\infty\). Its complete position–memory record \(G_\rho(q)\) retains both the position probability and the conditional quantum state of the memory. For every integer \(L\ge0\) and \(0<a\le1\), there is a state \(\omega\) satisfying \[G_\omega=G_\rho,\qquad \operatorname{Tr}(N\omega_A)=E,\qquad \operatorname{rank}\omega\le b(L+2),\qquad \tfrac12\|\rho-\omega\|_1\le\frac{E}{L+1}+a.\] Thus the approximants converge inside the same exact-record, exact-energy constraint set. A purification needs a new register of dimension at most \(b(L+2)\); retaining both the original memory and this register costs total dimension at most \(b^2(L+2)\). The construction preserves a positive head of the original state and replaces its actual positive tail. If the head has rank at most \(r\) and the tail mass is \(p\), at most \(b\) replacement vectors suffice, giving rank at most \(r+b\) and half trace-distance error at most \(p\) before energy restoration. A position-dependent unitary with \(\|U-I\|\le a\) then restores the exact energy without changing rank, spectrum or the measured record. With natural logarithms, \[|S(\rho)-S(\omega)|\le h_2(p)+p\,[g(E/p)+\log b],\] where \(h_2\) is binary entropy, \(g(x)=(1+x)\log(1+x)-x\log x\), and the zero-tail case is understood continuously. The same estimate controls quantum conditional entropy. Horizontal matrix purification removes kinetic overhead while handling zeros, changes of rank and the full real line. Scalar position distributions admit the corresponding construction for any fixed finite number of modes. The approximation is chosen from the known source, and finite rank refers to full wavefunctions rather than finite photon support. The integrated 12-page PDF contains two figures, complete proofs, and attribution of the classical kinetic, purification and phase identities used.
Can a finite-rank quantum state approximate a given optical state while reproducing an entire continuous measurement record and its original mean energy exactly? This paper provides a constructive affirmative answer, with quantitative rank, trace-distance and entropy control. The main result concerns one oscillator (A=L^2(\mathbb R)), with ([Q,P]=i) and photon operator (N=(Q^2+P^2-1)/2), coupled to a finite memory (B=\mathbb C^b). The input is any normal state (\rho_{AB}) with mean photons (E=\operatorname{Tr}(N\rho_A)<\infty). Its complete position–memory record (G_\rho(q)) retains both the position probability and the conditional quantum state of the memory. For every integer (L\ge0) and (0<a\le1), there is a state (\omega) satisfying [G_\omega=G_\rho,\qquad \operatorname{Tr}(N\omega_A)=E,\qquad \operatorname{rank}\omega\le b(L+2),\qquad \tfrac12|\rho-\omega|_1\le\frac{E}{L+1}+a.] Thus the approximants converge inside the same exact-record, exact-energy constraint set. A purification needs a new register of dimension at most (b(L+2)); retaining both the original memory and this register costs total dimension at most (b^2(L+2)). The construction preserves a positive head of the original state and replaces its actual positive tail. If the head has rank at most (r) and the tail mass is (p), at most (b) replacement vectors suffice, giving rank at most (r+b) and half trace-distance error at most (p) before energy restoration. A position-dependent unitary with (|U-I|\le a) then restores the exact energy without changing rank, spectrum or the measured record. With natural logarithms, [|S(\rho)-S(\omega)|\le h_2(p)+p,[g(E/p)+\log b],] where (h_2) is binary entropy, (g(x)=(1+x)\log(1+x)-x\log x), and the zero-tail case is understood continuously. The same estimate controls quantum conditional entropy. Horizontal matrix purification removes kinetic overhead while handling zeros, changes of rank and the full real line. Scalar position distributions admit the corresponding construction for any fixed finite number of modes. The approximation is chosen from the known source, and finite rank refers to full wavefunctions rather than finite photon support. The integrated 12-page PDF contains two figures, complete proofs, and attribution of the classical kinetic, purification and phase identities used.
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 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 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.
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.
When an optical source reproduces the local records of a thermal state, how much microscopic mixedness do those records actually guarantee? This paper gives explicit entropy bounds and a sharp large-budget law for that inference when the source's photon fluctuations are bounded. The setting is one oscillator with [Q,P]=i and N=(Q2+P2−1)/2. Fix a mean photon number E>0, and put v=E+1/2 and A=v−1/(4v). The complete position density, probability current and local kinetic density are required to be nv(q)=2πve−q2/(2v),jv(q)=0,kv(q)=(4v2q2+A)nv(q). These are exact functions, with no factor 1/2 in the definition of kv. They force mean photons E. The additional resource is the actual expectation Tr(N2ρ)≤M of the same source. At fixed mean, this is also a photon-number-variance cap. For every feasible finite M, every normal state compatible with these records satisfies S(ρ)≥h2(min{ηE(M),1/2})>0.ηE(M)=[M+M+A(A+B)A]2,B=1−2v1. Here S(ρ)=−Tr(ρlogρ), h2(t)=−tlogt−(1−t)log(1−t), and logarithms are natural. The bound applies to all compatible normal states, with no Gaussian or finite-rank restriction. Let Smin(E,M) be the entropy infimum over that full compatible class. As M→∞ at fixed E>0, Smin(E,M)=Mc(E)logM+c(E)[1−logc(E)]+OE(M2logM). The coefficient is c(E)=E2(E+1)2/(2E+1)2. A positive mixture of two pure Gaussian states preserves all three complete records and has an exactly computed photon-square cost. It supplies the matching upper bound, fixing both the leading coefficient and the next term. The same asymptotic law therefore holds within actual probability mixtures of pure Gaussian states. Finite-budget results are explicit bounds; the sharp equality is the stated fixed-E, large-M expansion. Without a photon-square cap, the exact compatible entropy range is (0,g(E)], where g(E)=(E+1)log(E+1)−ElogE. The thermal state uniquely attains the upper endpoint, while compatible Gaussian mixtures approach zero entropy. This separates a maximum-entropy model from the minimum entropy forced by the data. The proof uses established local Gram relations, projects onto an actual largest spectral component, and charges its positive spectral tail to the same source's N2 budget. This connection retains the column correlations needed for the lower bound. The analysis concerns one mode and ideal complete records; finite-precision or sampled records require additional error control. The integrated 13-page PDF contains two figures, complete proofs and references.
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.