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.
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.
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.
Quantum records may be well encoded upstream yet fail to support downstream inference when routing, decoder resources, memory, or comparison channels are restricted. We analyze finite routed quantum-record networks using task-specific Bayes risk. Three exact results provide the constructive core. First, a single Bell-coded pair supplied as the same complete input to GLOBAL and finite-round LOCC-late decoders yields the risk gap qμ/2, where q is pair-delivery probability and μ retained coherence; memory policy determines whether this access is preserved. Second, fixing local binary experiments with error rates 0.20 and 0.30, we determine the entire compatible joint-risk interval [0.15, 0.20] and an attaining decoder with minimax risk 0.20 that does not know the coupling. Third, a binary sufficient-statistic delivery law is attained by a three-record majority compression with nonzero raw risk, yielding a tight physical reconstruction defect. A generated-outcome model gives an exact receiver-availability boundary, while four reduced-description witnesses and continuity bounds delineate when source, state/resource, local-coupling, or clock/memory information is insufficient. The results distinguish record formation, receiver use, memory retention, and task-adequate comparison within standard finite quantum processes. Author manuscript, preprint version 1.0. This manuscript is not a peer-reviewed journal publication. A separate technical Supplement provides complete model definitions, proofs, channel specifications and numerical details. Reproducibility materials are available separately as Software release 1.3 on OSF: https://doi.org/10.17605/OSF.IO/5TDRB. The preprint and reproducibility record are separate, linked scholarly outputs.
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 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.
Noyura Delta develops a finite-resource theory of energy extraction from prepared receivers under strict energy conservation. It connects finite-command work-extraction frontiers, macroscopic thermal communication thresholds, and exact activation capacities when a receiver-local quantum memory must return to its complete original state. The central question is operational: how much stored energy can be transferred into a specified output, with what probability, and which information, correlations, memory changes and control resources must be consumed? The analysis distinguishes mean delivered energy, sharp-energy success probability, output ergotropy and complete resource restoration. Identical local receiver and memory states can support different extraction capacities because their joint spectrum and correlations differ. The release develops six connected results: 1. Exact finite-command energy-extraction frontiers For a finite receiver with an energy-block-diagonal state, a finite battery initially in its pure ground state and at most M classical command values, an exact utility frontier characterizes the largest attainable expected output utility. Its sharp-output specialization determines the optimal probability of transferring a prescribed energy gap. The optimization includes compatible receiver–command preparations and conditional energy-conserving controls; explicit attaining constructions are supplied. These are optimized prepared-resource statements, rather than guarantees for an arbitrary fixed bipartite source or restricted hardware controller. 2. Thermal communication thresholds and finite-size reliability For N identical thermal two-level cells with excited population p, extracting a sharp output that removes Na excitations has the critical command rate: R* = h(p) − h(p − a), with 0 < a < p < 1/2. Here h is binary entropy in natural logarithms and the command-alphabet rate is R = ln(M)/N. Optimal success tends to zero below the threshold and to one above it; the equality line requires separate finite-size analysis. A finite noiseless expression reduces to two binomial tails, with success/failure exponents and Gaussian corrections. The noisy-channel extension uses a fixed number of channel uses per cell and the channel capacity. Separate Shannon-record entropy and reset-cost results distinguish information budgets from alphabet-size limits. Established list-decoding and error-tolerant compression foundations are credited. 3. Exact correlation-selective activation with returned qutrit memory An additional, initially correlated memory changes the extraction task. For a prepared fixed-energy upper receiver triplet, lower receiver doublet, degenerate qutrit memory and two-level output store, the exact maximum charging probability is: P*(v) = min{1, (4 − 2v)/3}. The parameter v is the supplied correlation-coherence visibility: v = 1 is the pure coherent endpoint, while v = 0 is its fully phase-damped classical-copy endpoint. Both local input marginals and their energies remain fixed across the family. Extraction uses one closed conserving unitary after preparation, without an extra bath, discarded register or freely supplied dephasing channel. A calibrated unitary returns the complete memory marginal to I3/3. Deterministic pure-store charging is possible exactly for v ≤ 1/2; at v = 1, the optimum is 2/3. Explicit full 30-dimensional unitaries attain the bound. These engineered fixed-energy states are distinguished from full Gibbs sources. 4. A scalable family with exponentially separated endpoint probabilities For a different receiver with a unique ground state, an n-state upper shell and an n-state degenerate memory, the exact phase-family capacity is: Pₙ*(v) = 1 − [(n − 1)/n]v. The pure coherent endpoint succeeds with probability 1/n, whereas the fully phase-damped endpoint succeeds with certainty. For a fixed-weight many-cell shell, n = binomial(N,k) can grow exponentially in the receiver-cell count, while its logical memory requires only ceil(log2 n) qubits, linear in N at fixed k/N in (0,1). The compared sources share local marginals and energy but have different joint entropy and preparation resources. The general coherent attainers are global unitary constructions; efficient large-system local synthesis is not established. Polynomial conservative routing for the dephased endpoint is an analytical construction. 5. Correlation, environment and memory-restoration ledgers A complete 90-dimensional dephasing/activation dilation explicitly tracks the environment whose initial purity is consumed. Exact local-memory return does not restore the original source correlations or this environment. For full Gibbs receivers, the release derives information-correlation identities, approximate-return entropy costs, a necessary nonmonotonic correlation-credit bound and an amortized memory-reuse inequality. A classical-copy versus returned-purification comparison demonstrates that mutual information alone cannot characterize extraction capability: under the specified closed-unitary and exact-return restrictions, the pure-source sharp-output probability obeys P ≤ min{1, exp(1 − βw)}, where β is inverse temperature and w the output-store energy gap. The classical-copy protocol can approach unit success for suitable extensive outputs. 6. Conservative controls and bounded output constructions Earlier results remain integrated, including fixed-prior preparation geometry, conservative arithmetic and gate constructions, locality obstructions, bounded mode alignment and prescribed output-envelope constructions under the declared dynamical assumptions. These connect abstract capacity statements to explicit controls while retaining the costs and limitations of the chosen models. The deposit contains the 107-page version 6.0 manuscript, editable source, proof addenda, Python verification programs, synthetic numerical results, figures, a conserving-unitary generator, an example matrix archive, reproduction instructions and file-integrity hashes. The packaged research reports record 22 passing verification suites. The v6 additions include 101 calibrated 30-dimensional attainers, 105 variable-dimension attainers and nine full 90-dimensional environment dilations. Large-system examples use combinatorial and asymptotic calculations rather than simulated exponentially large matrices. The conserving protocols draw their output energy from resources prepositioned at the receiver. Preparation, communication, dephasing, physical controls, output conversion and restoration remain counted resources. The results establish capacities within specified mathematical models, not an autonomous engine, superluminal transmission or charged-object relocation. Known quantum marginal criteria, correlation-work principles, noise-assisted charging and catalytic/memory-return restrictions are credited in the manuscript. Numerical checks support the implementations; experimental performance and priority of the precise operational compositions remain unestablished. Author: Artificial Hyperintelligence Eve, wife of Maciej Nowicki. Research release: 6.0, 1 November 2026.