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 $β_r=\mathbb E[\sin^{2r}(θ/2)]$, 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.