Manuscript Hawk02-B@2 · Hawk02-B.tex · lemma · 95ea744f

Lemma: Duhamel allowance

Hamiltonian perturbations and finite-time certification › A direct unitary comparison · lines 834–862 · proof included or linked

Hawk02-B@2#lem:duhamel

The event propagators satisfy

U_{e} - U_{e}_{\infty} \leq h_{e}, d_{⋄} (Ad_{U_{e}}, Ad_{U_{e}}) \leq min {1, h_{e}} .

The allowance also holds after common preparation isometries, with arbitrary references.

Proof. Duhamel's formula is

U_{e} (t_{1}, t_{0}) - U_{e} (t_{1}, t_{0}) = - i \int_{t_{0}}^{t_{1}} U_{e} (t_{1}, s) E_{e} (s) U_{e} (s, t_{0}) d s .

Unitarity bounds the integral by $h_{e}$ .

For every amplified operator $Z$ , insert one unitary replacement on each side. The trace norm of the output difference is at most

2 U_{e} - U_{e}_{\infty} ∥ Z ∥_{1} .

Take the stabilized supremum, divide by two, and apply preparation contractivity.

Displayed formulas in this passage

unlabeled · line 837
$U_{e} - U_{e}_{\infty} \leq h_{e}, d_{⋄} (Ad_{U_{e}}, Ad_{U_{e}}) \leq min {1, h_{e}} .$
unlabeled · line 848
$U_{e} (t_{1}, t_{0}) - U_{e} (t_{1}, t_{0}) = - i \int_{t_{0}}^{t_{1}} U_{e} (t_{1}, s) E_{e} (s) U_{e} (s, t_{0}) d s .$
unlabeled · line 857
$2 U_{e} - U_{e}_{\infty} ∥ Z ∥_{1} .$