Manuscript En01-C@1 · En01-C.tex · section · 00a2161c

Fourth moments and propagation errors

Restricted reference computation and error ledger › Fourth moments and propagation errors · lines 1859–1947

En01-C@1#tab:errors

During quadratic Hamiltonian evolution, linear Gaussian diffusion, and regularized kicks, polynomial moments through degree four form a finite triangular system. For efficient computation during sensing, use Weyl-ordered moments

m_{j}^{(4)} = ⟨ x^{4 - j} p^{j} ⟩_{Weyl}, j = 0, \dots, 4.

The Hamiltonian contribution is

m_{j} ˙^{(4)} = \frac{4 - j}{M} m_{j + 1}^{(4)} - j κ m_{j - 1}^{(4)} + noise terms .

Quadratic monitoring contributes

4 μ ℏ^{2} j (j - 1) m_{j - 2}^{(4)} .

Gaussian diffusion contributes

ℏ^{2} D j (j - 1) ⟨ x^{4 - j} p^{j - 2} ⟩_{Weyl} .

The regularized kick contribution has the same second-order term with $D = λ A_{r_{c}}$ , plus, for $j = 4$ ,

λ ℏ^{4} \int k_{e}^{4} Q_{r_{c}} (d k) .

Terms with out-of-range indices are omitted. These identities follow by expanding a momentum shift and by the exact quadratic Wigner commutator. They permit a uniform fourth-moment enclosure once the initial moments, parameter ranges, and integration error have been bounded. They do not by themselves supply those numerical enclosures.

For full-state approximations, if an admitted exact propagator is compared with an approximate Hermitian path $ρ$ on a common variation-of-constants domain, the residual

r (t) = ρ ˙ (t) - L (t) ρ (t)

gives

T (ρ (t), ρ (t)) \leq T (ρ (0), ρ (0)) + \frac{1}{2} \int_{0}^{t} ∥ r (u) ∥_{1} d u .

This follows by propagating the initial difference and residual and using CPTP contraction. The same estimate applies to complete direct-sum instrument blocks.

Application departure	Required accounting
Cooling/reset uncertainty	Shared setting law and calibrated moment region
Nonflat free potential	Derived curvature/noise response and timing uncertainty
Colored support spectrum	Cross-stage kernel, not a substituted resonant coefficient
Gas and magnetic forces	Independent aggregate free-diffusion constraints
Finite cavity elimination	Complete sensing-instrument or relevant moment error
Setting-dependent measurement strength	Physical-coordinate transfer calibration
Split record saturation or loss	Clipping model and recorded missing-outcome law
Finite packet and window averaging	Width, phase, population imbalance, and tail bounds
Regularized continuum calculation	Spatial/momentum tails, time error, kick quadrature, and residuals
Statistical calibration	Joint coverage and independence or valid simultaneous construction

needed before the conditional certificate becomes an apparatus-specific sensitivity claim.

Displayed formulas in this passage

unlabeled · line 1865
$m_{j}^{(4)} = ⟨ x^{4 - j} p^{j} ⟩_{Weyl}, j = 0, \dots, 4.$
unlabeled · line 1870
$m_{j} ˙^{(4)} = \frac{4 - j}{M} m_{j + 1}^{(4)} - j κ m_{j - 1}^{(4)} + noise terms .$
unlabeled · line 1877
$4 μ ℏ^{2} j (j - 1) m_{j - 2}^{(4)} .$
unlabeled · line 1881
$ℏ^{2} D j (j - 1) ⟨ x^{4 - j} p^{j - 2} ⟩_{Weyl} .$
unlabeled · line 1887
$λ ℏ^{4} \int k_{e}^{4} Q_{r_{c}} (d k) .$
unlabeled · line 1900
$r (t) = ρ ˙ (t) - L (t) ρ (t)$
unlabeled · line 1904
$T (ρ (t), ρ (t)) \leq T (ρ (0), ρ (0)) + \frac{1}{2} \int_{0}^{t} ∥ r (u) ∥_{1} d u .$