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

Integrated residual certificates

Conditional approximation and verification interfaces › Integrated residual certificates · lines 2974–2996

En03-A@1#ip:residual, En03-A@1#eq:integrated-residual

Let $U (t, s)$ be a finite-dimensional linear propagator contractive in trace norm on Hermitian operators. Suppose

σ ˙ (t) = L_{t} σ (t), σ ˙ (t) = L_{t} σ (t) + r (t),

where $r (t)$ is integrable and Hermitian and the usual variation-of-constants identity holds. Then

∥ σ (t) - σ (t) ∥_{1} \leq ∥ σ (0) - σ (0) ∥_{1} + \int_{0}^{t} ∥ r (u) ∥_{1} d u .

This applies to unnormalized instrument blocks when their declared propagators are CP and trace nonincreasing.

Displayed formulas in this passage

En03-A@1#eq:integrated-residual · line 2987
$∥ σ (t) - σ (t) ∥_{1} \leq ∥ σ (0) - σ (0) ∥_{1} + \int_{0}^{t} ∥ r (u) ∥_{1} d u .$
unlabeled · line 2981
$σ ˙ (t) = L_{t} σ (t), σ ˙ (t) = L_{t} σ (t) + r (t),$