Manuscript En01-B@2 · En01-B-v2.tex · proposition · f226d1eb

Proposition: State-specific residual certificate

Near-obstruction in the full regularized model · lines 1451–1473 · proof included or linked

En01-B@2#prop:residual

Let $ρ (t)$ be an admitted exact evolution and $ρ (t)$ a Hermitian normalized absolutely continuous approximation on a common variation-of-constants domain. If

r (t) = ρ ˙ (t) - L (t) ρ (t)

is integrable in trace norm, then

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

The statement also applies to direct sums of unnormalized instrument blocks.

Proof. Propagate the initial difference and the residual integral. The admitted propagator contracts Hermitian trace norm.

Displayed formulas in this passage

unlabeled · line 1457
$r (t) = ρ ˙ (t) - L (t) ρ (t)$
unlabeled · line 1461
$T (ρ (t), ρ (t)) \leq T (ρ (0), ρ (0)) + \frac{1}{2} \int_{0}^{t} ∥ r (u) ∥_{1} d u .$

Nearby in this section

proposition: Proposition: Bounded-aperture generator remainder
theorem: Theorem: Full-model near-obstruction certificate
section: Near-obstruction in the full regularized model