Manuscript En03-A@1 · En03-A.tex · proof · 5398614e

Proof

Conditional approximation and verification interfaces › Integrated residual certificates · lines 2997–3011

Proof. The difference is

U (t, 0) (σ (0) - σ (0)) + \int_{0}^{t} U (t, u) r (u) d u .

Contractivity and the triangle inequality give the result. For a CP trace-nonincreasing map and Hermitian $H = H_{+} - H_{-}$ , positivity gives

∥ U (H) ∥_{1} \leq Tr U (H_{+}) + Tr U (H_{-}) \leq ∥ H ∥_{1} .

Displayed formulas in this passage

unlabeled · line 2999
$U (t, 0) (σ (0) - σ (0)) + \int_{0}^{t} U (t, u) r (u) d u .$
unlabeled · line 3006
$∥ U (H) ∥_{1} \leq Tr U (H_{+}) + Tr U (H_{-}) \leq ∥ H ∥_{1} .$