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.

The paper studies a fixed finite-dimensional quantum encoding exposed to $e^{-i\sigma xH}$, with known $H$ but unobserved random angle, followed by one deterministic recovery. It asks how encoding geometry and incomplete calibration jointly limit recovery fidelity. For a nondegenerate prefix correcting $I,H,\ldots,H^L$, it derives the exact leading optimal entanglement infidelity, $1-F_{e,\mathrm{opt}}=g_L^2h_m\mathcal V\sigma^{2m}/(m!)^2+o(\sigma^{2m})$, where $m=L+1$. Physical block-Krylov orthogonalization determines propagation $g_L$ and logical boundary distortion $\mathcal V$; projection of $x^m$ onto lower angle powers determines residual variance $h_m$. A coherently adjusted, polar-aligned recovery attains the coefficient, with a converse covering all CPTP recoveries. After a known logical rotation is removed, optimal half-diamond error lies between one and $d$ times optimal entanglement infidelity at leading order. Petz recovery doubles the occupied-shell contribution but retains the optimal leakage contribution. For uncertain priors, a minimax polynomial selects one common decoder. Commensurate spectral and calibration frequencies permit exact finite moment optimization. With sole datum $\beta_r=\mathbb E[\sin^{2r}(\theta/2)]$, worst-prior infidelity is $O(\beta_r)$ exactly when powers through $r-1$ are correctable; a first defective power $m<r$ instead gives $\beta_r^{m/r}$ scaling under additional boundary hypotheses. In a loss-free four-photon example, two calibration durations retain the informed quartic coefficient, whereas one permits quadratic loss. The local theorem requires rank and moment conditions; uniform guarantees require tail control. The model excludes independently fluctuating or noncommuting generators and control during storage. Optical recovery implementability and photon loss remain unresolved.