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Finite-classical-record quantum state reconstruction: one-shot bounds and acquisition limits

Wahyu Rahmaniar

Published Sep 24, 2026• 0 citations
Publisher

Abstract

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

DOI
10.21203/rs.3.rs-10997700/v1

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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
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Free classical communication can yield an unbounded single-shot saving in quantum compression, even though it leaves the asymptotic rate of an i.i.d. source unchanged. At every fixed error in (0,1)(0,1)(0,1), we exhibit sources whose unassisted compression cost is arbitrarily large, yet a single qubit suffices with classical communication and no preshared entanglement. We establish this separation by determining the exact optimal source fidelity for compression through an mmm-dimensional noiseless quantum channel, with and without free classical communication, while preserving correlations with an inaccessible reference. Without classical assistance, the optimal fidelity is exactly the Ky Fan norm ∥ρ∥(m)\|ρ\|_{(m)}∥ρ∥(m)​, the sum of the mmm largest eigenvalues of the source state. With classical assistance, we obtain an explicit spectral threshold formula and a protocol attaining it. Classical communication strictly improves fidelity whenever mmm is smaller than the source rank. The corresponding compression costs are two smoothed versions of the log-rank entropy, also called max-entropy: trace-distance smoothing under local operations and quadratic smoothing under LOCC. For i.i.d. copies of a fixed source with positive entropy variance, however, the two costs have identical first-, second-, and third-order asymptotic terms. For nonlattice spectra, we determine the constant-order advantage: classical communication saves log⁡2(e)−1\log_2(e)-1log2​(e)−1 bits per block asymptotically, and the ratio of assisted to unassisted minimum communication dimensions converges to 2/e2/e2/e.
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 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.
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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.