Krylov geometry and optimal recovery under uncertain coherent noise
Abstract
Recovery from a coherent rotation with an unobserved angle depends on both the encoding and what calibration reveals about the noise. We determine their combined limit for a known finite-dimensional generator and one final recovery. Two polynomial projections organize the answer. A physical Krylov chain separates correctable error amplitudes from their first state-dependent distortion; projection onto lower-angle powers identifies the part that a coherent decoder can cancel. Under explicit rank and moment conditions, their combination gives the exact leading optimal entanglement infidelity at weak noise and a recovery attaining it for any finite logical dimension. The optimal worst-input error lies between one and d times that infidelity at leading order, for logical dimension d. When the angle distribution is unknown, we optimize one decoder for every distribution compatible with the calibration. When spectral and calibration frequencies are integer multiples of a common frequency, moment positivity turns this into an exact finite optimization. Small measured means can still allow rare finite rotations. If the sole calibration datum is βr=𝔼[sin2r(θ/2)], with θ the wrapped rotation phase at that common frequency, the optimal worst-prior infidelity is O(βr) exactly when generator powers through degree r−1 are correctable. When the first uncorrectable power has degree m<r, the optimal loss scales as βrm/r, with a sharp coefficient fixed by a universal moment problem. In a loss-free four-photon memory, population measurements at two calibration durations retain the fully informed quartic error coefficient for a fixed symmetric narrow-noise family with positive variance and finite fourth moment; a single duration permits loss quadratic in the noise width.
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BibTeX
@misc{chowdhury2026krylov,
title = {{Krylov geometry and optimal recovery under uncertain coherent noise}},
author = {Abhishek Chowdhury},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.33607},
eprint = {2610.33607},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.33607},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Krylov geometry and optimal recovery under uncertain coherent noise
AU - Abhishek Chowdhury
PY - 2026
DA - 2026/10/24
PB - arXiv
DO - 10.48550/arxiv.2610.33607
UR - https://arxiv.org/abs/2610.33607
AB - Recovery from a coherent rotation with an unobserved angle depends on both the encoding and what calibration reveals about the noise. We determine their combined limit for a known finite-dimensional generator and one final recovery. Two polynomial projections organize the answer. A physical Krylov chain separates correctable error amplitudes from their first state-dependent distortion; projection onto lower-angle powers identifies the part that a coherent decoder can cancel. Under explicit rank and moment conditions, their combination gives the exact leading optimal entanglement infidelity at weak noise and a recovery attaining it for any finite logical dimension. The optimal worst-input error lies between one and d times that infidelity at leading order, for logical dimension d. When the angle distribution is unknown, we optimize one decoder for every distribution compatible with the calibration. When spectral and calibration frequencies are integer multiples of a common frequency, moment positivity turns this into an exact finite optimization. Small measured means can still allow rare finite rotations. If the sole calibration datum is β_r=𝔼[sin^2r(θ/2)], with θ the wrapped rotation phase at that common frequency, the optimal worst-prior infidelity is O(β_r) exactly when generator powers through degree r−1 are correctable. When the first uncorrectable power has degree m<r, the optimal loss scales as β_r^m/r, with a sharp coefficient fixed by a universal moment problem. In a loss-free four-photon memory, population measurements at two calibration durations retain the fully informed quartic error coefficient for a fixed symmetric narrow-noise family with positive variance and finite fourth moment; a single duration permits loss quadratic in the noise width.
ER -