Manuscript Hawk02-C@2 · Hawk02-C.tex · theorem · bb332d66

Theorem: Sector Duhamel allowance

Perturbations on the checked invariant sector · lines 1372–1417 · proof included or linked

Hawk02-C@2#thm:C6, Hawk02-C@2#eq:duhamelchannel

The bounded interaction-picture construction defines a unique unitary sector propagator. For

Δ_{T} = \int_{0}^{T} ∥ E_{1} (t) ∥_{\infty} d t,

it satisfies

U_{1} (T, 0) - e^{- i T H_{1}}_{\infty} \leq Δ_{T} .

On the common prepared code and identified complete outputs,

d_{⋄} (Ad_{W_{T}}, Ad_{W_{T}}) \leq min {1, Δ_{T}} .

The same allowance holds after ideal packet factorization and for either partial trace.

Proof. The interaction-picture perturbation is

E_{I} (t) = e^{i t H_{1}} E_{1} (t) e^{- i t H_{1}} .

It has the same norm and is strongly measurable. The Dyson integrals exist strongly; the $n$ th term is bounded in norm by $Δ_{T}^{n} / n!$ . The norm-convergent majorant gives existence and uniqueness. Hermiticity gives unitarity.

Variation of constants yields

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

Unitarity bounds the norm by $Δ_{T}$ . Compose with the common preparation and apply Theorem thm:C1.

Displayed formulas in this passage

Hawk02-C@2#eq:duhamelchannel · line 1386
$d_{⋄} (Ad_{W_{T}}, Ad_{W_{T}}) \leq min {1, Δ_{T}} .$
unlabeled · line 1377
$Δ_{T} = \int_{0}^{T} ∥ E_{1} (t) ∥_{\infty} d t,$
unlabeled · line 1381
$U_{1} (T, 0) - e^{- i T H_{1}}_{\infty} \leq Δ_{T} .$
unlabeled · line 1399
$E_{I} (t) = e^{i t H_{1}} E_{1} (t) e^{- i t H_{1}} .$
unlabeled · line 1409
$U_{1} (T, 0) - e^{- i T H_{1}} = - i \int_{0}^{T} U_{1} (T, s) E_{1} (s) e^{- i s H_{1}} d s .$

Theorem: Sector Duhamel allowance

Displayed formulas in this passage

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