How much quantum storage must a hidden implementation retain when experimental observations leave its structure undetermined? This paper establishes exact minimum quantum-storage requirements across all implementations compatible with specified channel observations. The resource is the worst-input expected logarithm of retained quantum dimension across one controlled temporal cut, with unlimited classical records available at no cost. For the channel dimension measure introduced by Cope and Uola, the paper proves their suggested minimax equality relating minimum storage to output Schmidt measure and extends it to every nonempty compact convex set of compatible channels. One common input enforces the resource minimum across all compatible implementations. A finite optimal implementation exists, although an optimal entangled probe need not. For qubit channels, four measured coefficients determine an exact, attained minimum without assuming channel covariance or unitality. An explicit five-Kraus construction attains 99/169, while a separate analytical certificate proves that every compatible implementation must use nontrivial quantum storage with at least that probability on the same specified input. A second exact formula uses only two Pauli-transfer coefficients and yields a minimum storage requirement of 0.58 when both coefficients are jointly constrained to the interval [0.79, 0.81]. The paper also presents semidefinite optimization over compatible experimental uncertainty regions. The results require calibrated quantum inputs and outputs, concealed preparation labels, and accounting for every quantum carrier across the storage cut. They constrain storage resources without reconstructing the hidden implementation or certifying environmental non-Markovianity. Motivated by Shadow Theory’s source/readout distinction, the proofs use standard quantum mechanics and quantum-information mathematics. Version 1.1 includes complete proofs, explicit attaining constructions, worked examples, and discussion of experimental assumptions and limitations. No experimental data are reported.
How much quantum storage must a hidden implementation retain when experimental observations leave its structure undetermined? This paper establishes exact minimum quantum-storage requirements across all implementations compatible with specified channel observations. The resource is the worst-input expected logarithm of retained quantum dimension across one controlled temporal cut, with unlimited classical records available at no cost. For the channel dimension measure introduced by Cope and Uola, the paper proves their suggested minimax equality relating minimum storage to output Schmidt measure and extends it to every nonempty compact convex set of compatible channels. One common input enforces the resource minimum across all compatible implementations. A finite optimal implementation exists, although an optimal entangled probe need not. For qubit channels, four measured coefficients determine an exact, attained minimum without assuming channel covariance or unitality. An explicit five-Kraus construction attains 99/169, while a separate analytical certificate proves that every compatible implementation must use nontrivial quantum storage with at least that probability on the same specified input. A second exact formula uses only two Pauli-transfer coefficients and yields a minimum storage requirement of 0.58 when both coefficients are jointly constrained to the interval [0.79, 0.81]. The paper also presents semidefinite optimization over compatible experimental uncertainty regions. The results require calibrated quantum inputs and outputs, concealed preparation labels, and accounting for every quantum carrier across the storage cut. They constrain storage resources without reconstructing the hidden implementation or certifying environmental non-Markovianity. Motivated by Shadow Theory’s source/readout distinction, the proofs use standard quantum mechanics and quantum-information mathematics. Version 1.1 includes complete proofs, explicit attaining constructions, worked examples, and discussion of experimental assumptions and limitations. No experimental data are reported.
How much internal quantum memory is needed to reproduce a stationary stochastic process, and how does reducing that memory change the environmental resources required when output records are retained? This paper establishes exact quantum-source constructions and resource bounds under explicitly specified readout and retained-output conditions. A general completion theorem identifies the minimum Hilbert-space dimension for stationary triple data in the orthogonal-current-symbol source class: it equals the sum of the complex positive-semidefinite ranks of the middle-symbol slices. An attaining construction reproduces the entire canonical stationary order-two process. For the delayed trine process, the minimum classical hidden Markov model requires nine states at every non-independent parameter value. Quantum instruments reproduce the same complete output law using four-dimensional memory throughout the parameter range, with four dimensions proved necessary in a certified low-noise regime and three dimensions both sufficient and necessary on an explicit high-noise interval. The paper then separates reusable memory dimension from environmental dilation cost. On the certified high-noise interval, the exact one-step Pareto frontier consists of a three-dimensional source with recorded-channel Choi rank nine and a four-dimensional source with rank three. Here, Choi rank measures the minimum initially pure environmental remainder dimension for the channel retaining both the emitted classical symbol and final quantum memory. A separate mixed-support qutrit construction at noise parameter 0.99 improves complete-history costs and gives upper bounds close to universal lower bounds. These results show why optimizing a single step need not optimize a retained multi-step history. The analysis also distinguishes complete channels, fixed-input states and archive-only outputs, and demonstrates how access to a retained reference can distinguish generators with identical passive output statistics. Full-word proofs, algebraic complete-positivity certificates, a technical companion and reproducibility materials accompany the results. The exact intermediate memory boundary and complete-history optimum remain open. All results use standard quantum probabilities and apply to the stated operational resource models; they do not establish total hardware, energy or thermodynamic minima.
Abstract Preparing additional quantum states while retaining a source requires information beyond an arbitrary unknown input. We formulate reconstruction from incom- plete classical preparation records as an identity channel on the source and two output preparations. For a short interval of QPE phase states, we derive the exact record-only minimax fidelity and an attaining endpoint superposition, closing the gap between center replay and an endpoint upper bound. An explicit one-ancilla circuit realizes the optimum with a known success probability, while deterministic synthesis provides a finite-circuit alternative. Shor and randomized-simulation examples expose different obstructions from discrete ambiguity and missing ran- dom choices. In one randomized-simulation setting, optimal conditional fidelity averaged over records was 0.925, although robust fidelity above five sixths was impossible for eight of ten records. Session calibration and resource-controlled comparisons distinguish average performance from worst-case guarantees. Hard- ware diagnostics test operational phase models, source waiting and preparation costs. They do not certify intrinsic gate angles, reconstruction fidelity or physical source preservation. The analysis identifies information and operational conditions without changing universal cloning bounds.
A finite quantum memory is specified by more than its entanglement entropy. This paper determines how to shape its Schmidt weights and allocate optical photons to minimize the quantum conditional uncertainty of a prescribed heterodyne measurement. The resulting resource laws let researchers compare a uniformly weighted entangled source with a shaped source, and assess when sharing one memory across optical modes is more efficient than independently preparing mode–memory pairs. Complete preparation model and sharp leading resource law. Let Vm,D(E) be the minimum of Hμ(Z∣B)=hμ(Z)−χ(Z:B) over all normal, possibly mixed optical–memory states with total mean photon number at most E and total accessible memory dimension at most D. Here Z is the outcome of an m-mode heterodyne measurement, B is the retained quantum memory, χ is Holevo information, and dμ=π−md2mz. Accessible flags count toward the same memory budget. For every fixed m and every joint limit D,E→∞, Vm,D(E)∼TDΛm(E/TD),TD=(m!D)1/m. The function Λm is an explicitly characterized constrained radial ground-state energy. Its Kummer and Bessel profiles give legal attaining preparations. The converse retains arbitrary coherent inputs and their photon tails, rather than optimizing only a diagonal ansatz. Why shaping changes the resource exponent. At the common photon ceiling E=TD, optimally shaped sources have uncertainty of order D−1/m, whereas final flat Schmidt spectra have optimal order D−1/(2m), even after optimizing the occupied optical subspace. Flat weights create a sharp support boundary; tapering the weights replaces its boundary-layer cost by a smaller gradient cost. To reach a small target uncertainty ε, this changes the leading memory requirement from 2mlog2(1/ε) to mlog2(1/ε), and the attainable photon scaling from quadratic to linear in 1/ε. This improvement is compatible with nearly maximal entanglement entropy. For the explicit canonical saturated construction, S(ρB,D)=lnD−cm+o(1),∥ρB,D−ID/D∥1⟶Lm>0. The full trace-norm limits are approximately L1=0.8771 and L2=1.1170; the half-trace-distance convention halves these numbers. Thus a vanishing relative entropy deficit does not make the optimized spectrum uniform. Collective preparation at fully counted resources. For two modes in the energy-sufficient asymptotic regime, sharing quantum memory reduces the leading conditional entropy by about 10.24% relative to the optimized independent mode–memory preparation class. That comparator includes local state optimization, integer memory and energy allocations, mixtures and charged accessible flags; it is not the class of all separable joint states. A separate finite construction uses seven coherent optical inputs, three retained qubits and 0.21 input photons. Under the specified loss, phase and stochastic memory-fault contract, its retained photon budget is at most 0.200781, and its conditional entropy is strictly lower than that of the same independent comparator by more than 0.00547535 nats. The integrated article gives the preparation circuit, one-sided numerical bounds and complete same-source error accounting. Interpretation. The objective uses quantum side information. A classical measurement of the memory generally has accessible information below χ(Z:B), so the conditional-entropy advantages do not by themselves assert the same improvement in estimation precision or classical readout performance. The finite circuit is a rigorously specified source model, not a qualification of an implemented laboratory apparatus. The single PDF contains the article, five figures, full 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.
Quantum illumination uses a retained idler to improve detection of a weak optical return in thermal noise. For a source designer, the practical resource question is how much ideal quantum storage is needed for a specified improvement. This work gives the exact asymptotic answer when the signal-photon budget and the total retained-register dimension are constrained independently. Exact finite-memory frontier Consider a thermal attenuator with transmissivity 0<η<1 and vacuum-output background occupation b>0, tested against the replacer that returns the same thermal state. For n parallel uses, every admissible probe obeys a mean total signal-photon ceiling nN and a strict bound dimR≤⌊enP⌋ on the entire retained register, including retained source records. The memory rate is P/ln2 ideal qubits per use. With vanishing miss probability, the optimal false-alarm exponent is Ξ(N,P)=ηln(1+1/b)N+G(min{P,g(N)}). Here all logarithms are natural, g(x)=(x+1)ln(x+1)−xlnx, with g(0)=0, and G(p)=g(b)+p−F(g−1(p)). The function F(x) is the entropy of the full complementary output for a thermal input of mean occupation x; the complement includes the purification of the thermal environment. The first term in Ξ is the optimum without retained memory, and G is the additional detection benefit. The converse covers arbitrary normal mixed probes, correlations across channel uses and collective measurements. Matching sources satisfy the actual finite-register and energy bounds: they allocate photons between signal–idler correlations and a known displacement, and approach the frontier through finite-support preparations. How much storage is useful? The exact saturation rate and initial marginal value are Psat(N)=g(N),G′(0+)=1+bη. Because G is strictly increasing and strictly concave, at a positive signal budget N>0 the minimum rate retaining a fixed fraction 0<r<1 of the full additive memory benefit is Pr(N)=G−1(rG(g(N)))<rg(N). At low brightness, Pr(N)∼rg(N). At high brightness, this fixed-fraction storage cost remains bounded while the exact-saturation cost diverges. The relative detection benefit simultaneously tends to zero: saving a large fraction of the saturation storage does not imply a large improvement over the total no-memory exponent. Connection to optical communication For the same thermal channel, let C(N,P) denote the classical capacity under the corresponding photon budget and a strict initial maximally entangled Schmidt-rank supply. The two separately defined tasks satisfy Ξ(N,P)−Ξ(N,0)=C(N,P)−C(N,0). This identity equates optimized incremental values, not the physical resources or protocols. The manuscript explicitly attributes the shared Gaussian-reference contraction and entropy-support tools to the capacity analysis. Its detection-specific contribution is the original-register mixed-probe converse, strict finite-register attainment and operational testing limit. Scope. The result concerns parallel, phase-known thermal attenuation, ideal retained storage and the vanishing-miss Stein criterion. It is an asymptotic resource benchmark rather than a finite-sample receiver or a noisy-memory optimum. The integrated article includes the complete proofs and two figure groups showing both storage cost and the value obtained from it.
How 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.