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

Proposition: Event operator-norm formula

Scale, records, and residual instruments › Exact unassisted error · lines 2376–2399 · proof included or linked

En03-A@1#prop:recorderror, En03-A@1#eq:delta-exact

With eq:distance-conventions,

δ_{rec} = A \subseteq W max w \in A \sum D_{w}_{\infty} \leq \frac{1}{2} w \sum ∥ D_{w} ∥_{\infty} .

Proof. For real zero-sum $d_{w}$ ,

\frac{1}{2} w \sum ∣ d_{w} ∣ = A \subseteq W max ∣ w \in A \sum d_{w} ∣ .

Use $d_{w} = Tr (ρ D_{w})$ , interchange the finite maximum with the state supremum, and apply $sup_{ρ} ∣ Tr (ρ H)∣ = ∥ H ∥_{\infty}$ for Hermitian $H$ . The upper bound follows termwise.

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En03-A@1#eq:delta-exact · line 2379
$δ_{rec} = A \subseteq W max w \in A \sum D_{w}_{\infty} \leq \frac{1}{2} w \sum ∥ D_{w} ∥_{\infty} .$
unlabeled · line 2391
$\frac{1}{2} w \sum ∣ d_{w} ∣ = A \subseteq W max ∣ w \in A \sum d_{w} ∣ .$

Proposition: Event operator-norm formula

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