Manuscript En01-A@1 · En01-A.tex · lemma · 558b972b

Lemma: Telescoping state error

Robustness and the instability of exact order · lines 2507–2535 · proof included or linked

En01-A@1#lem:telescope

Let

ρ = G_{L} \dots G_{1} (ρ_{0}), ρ ~ = G_{L} \dots G_{1} (ρ_{0}),

where all maps are CPTP and

∥ G_{ℓ} - G_{ℓ} ∥_{⋄} \leq ε_{ℓ} .

Then

∥ ρ ~ - ρ ∥_{1} \leq ϵ, ϵ = ℓ = 1 \sum^{L} ε_{ℓ} .

Proof. Expand the difference of the compositions as a telescoping sum, using ideal maps before the differing factor and perturbed maps after it. The differing factor contributes at most $ε_{ℓ}$ in trace norm. Subsequent CPTP maps contract the Hermitian difference. Sum the bounds.

Displayed formulas in this passage

unlabeled · line 2509
$ρ = G_{L} \dots G_{1} (ρ_{0}), ρ ~ = G_{L} \dots G_{1} (ρ_{0}),$
unlabeled · line 2516
$∥ G_{ℓ} - G_{ℓ} ∥_{⋄} \leq ε_{ℓ} .$
unlabeled · line 2521
$∥ ρ ~ - ρ ∥_{1} \leq ϵ, ϵ = ℓ = 1 \sum^{L} ε_{ℓ} .$

Lemma: Telescoping state error

Displayed formulas in this passage

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