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.
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.
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.
Abstract We study the one-shot reconstruction of a quantum state when a single copy of the source is measured, the outcome is stored in a classical record of fixed finite dimension, and a state is prepared at the destination from the record. This acquisition-limited problem is distinguished from a visible benchmark in which the source label is available to the encoder. For a finite ensemble we derive finite-record bounds on the visible distortion in terms of the cumulative prior mass covered by the record and the optimal discrimination probability, identify the regimes in which they are exact, and solve the two-state visible problem at every record dimension. For pure-state ensembles, exact acquisition-limited reconstruction is possible if and only if the states are perfectly distinguishable, so nonorthogonal targets retain a positive error at every record dimension. For two equiprobable nonorthogonal pure states we derive the optimal trace-distance distortion within the reflection-symmetric two-outcome class in closed variational form and find that unrestricted numerical optimization with up to six outcomes attains the same value; the resulting symmetric value lies strictly below the guess-and-prepare, unambiguous-discrimination, fidelity-optimal, and Haar-random strategies and is realized by one classical bit and one feed-forward rotation. Every classical-record reconstruction is shown to be entanglement breaking, with a quantitative bound on the loss of entanglement with an inaccessible reference, and the record dimension is related to the spectral rank of an energy-constrained memory. These results quantify the separation between a finite classical description and the information obtainable from a single quantum system. MSC Classification: 81P15 , 81P17 , 81P45 , 81P47
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.
This preprint studies operational equivalence and recovery in a twenty-dimensional residual system with a fixed four-sector representation of dimensions 8, 8, 2 and 2. It distinguishes what a declared observation algebra can detect, what a channel preserves, and what a retained classical record makes recoverable. Record-resolved criteria separate useful reversible-error records from unrelated labels and measured-sector records, even when the discarded channel is the same. For a specified coarse-record model, the work identifies exactly recoverable observable algebras and proves an optimal entanglement-fidelity expression on a fixed four-state orbit code. Preparation-aware recovery is then classified over all one-use isometric subspaces in the stated family. An explicit ten-dimensional marker code attains exact recovery under a specified stochastic record where the whole-sector support restriction permits only eight dimensions. Its encoding, nonbaseline marker measurement, and conditional inverse operations remain supplied resources. Scoped Lean evidence verifies the explicit arbitrary-complex-matrix recovery construction and related finite identities; it does not formalize the general complete-positivity/reference framework or optimal-dimension theorem. The recorded-error model is not derived from the companion exchange-timing model, and no protection of its long pulse sequence is claimed.
A calibrated quantum-record test can require both a global target state and a commoncomplete reader input, neither of which is certified by accessible marginal data. We construct theseinputs for the specified source-tagged model at an independently known source length m= 65. Thepublished joint-input work/source reset is followed by 37 finite swaps with fresh reader qubits and a69-slot target sequence: coherent work-label creation, globally conditioned tag injection, and 64neighboring source transfers. The control-only sequence prepares the exact far-target cat or itssigned partner with the required common phase. A retained random sign produces the comparisonmixture without postselection. Full-state comparisons include consumed supplies and arbitraryinitial correlations with exactly inert retained memory. The original clock, holding terms, sourcehopping, both port directions and either actual law, h= 0 or h= J/25, remain active throughout.The constructed initialization fits the existing 1/4000 allowance; target delivery has sufficienterror below 0.004816692 <1/200. With separate complete-interval generator and endpoint-effectallowances, each target response differs from its nominal value by less than 0.007816692 <1/125.The same reader initialization supplies the published reference-preparation interface. The resultis a conditional single-delivery construction: it consumes 37 fresh pure reader qubits, requirespreparation access through site 65 and additional globally conditioned controls, and inherits anextremely costly reset. Application to repeated acquisition still requires almost-sure conditionalsupply and error bounds; practical feasibility and uniform passing power are not established.
A quantum record can have locally accessible populations while its distinguishing phase remains correlated with a distant source. We study this access boundary in a specified finite recording apparatus with a returning, vacuum/one-excitation source. Source-position weights give finite-time cutoff bounds without removing any physical hopping or return coupling. For an initially far-tagged record and its incoherent mixture, every reader in a declared position-preserving local-control class obeys a phase-visibility envelope and a necessary access-time inequality. A separate finite local probe shows why that state-discrimination limit is not an unrestricted limit on distinguishing two dynamical laws: fresh local coherence can reveal a stipulated added dephaser without recovering the old record phase. We construct a bounded, degree-three reader accessing only source positions 1–32. With the initial tag fixed at position 65, exact finite-source arrival certificates and an analytical tail comparison yield recovery uniformly over every finite total source length at least 65. Accessible control faults are charged against the arrived signed-phase norm, while source-moving drift is charged against the full norm. At the fixed deadline 25.02/J, the matched phase-dependent difference between coherent and archive-gated dynamics exceeds 0.005, including the stated simultaneous control faults and four additional absolute response allowances of 0.008 each. Complementary results cover a vacuum-ready source and a stronger endpoint-prepared finite example. The conclusions are conditional on the supplied model, preparations, common realized controls and calibration bounds. They are not a switchable-fundamental-law experiment, a nuisance-independent classifier, an optimal-speed theorem or an end-to-end formal verification. The supporting Lean companion contains 59 declarations in eight modules with accepted returned Mac compilation, type and axiom evidence; physical analytical interfaces remain explicit hypotheses or outside that subset. The deposit files include the main manuscript, technical supplement and full research/reproduction package.