Manuscript VII-E@1 · VII-E.tex · theorem · b713cf3c

Theorem: Event operator-norm formula

Record compatibility, residuals, and composition › The exact unassisted record error · lines 1168–1197 · proof included or linked

VII-E@1#thm:event-norm, VII-E@1#eq:event-norm

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With the conventions in eq:distance-conventions,

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

The maximum is attained by a pure input state.

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

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

Use $d_{w} = Tr (ρ D_{w})$ . Since the event set is finite,

δ_{rec} = A \subseteq W max ρ sup ∣ Tr (ρ D_{A})∣ .

For a Hermitian operator $H$ , the latter supremum is $∥ H ∥_{\infty}$ , attained at an eigenvector whose eigenvalue has maximal absolute value. This proves the equality and attainment. The bound follows from $∣ Tr (ρ D_{w})∣ \leq ∥ D_{w} ∥_{\infty}$ .

Displayed formulas in this passage

VII-E@1#eq:event-norm · line 1171
$δ_{rec} = A \subseteq W max ∥ D_{A} ∥_{\infty} \leq \frac{1}{2} w \sum ∥ D_{w} ∥_{\infty} .$
unlabeled · line 1183
$\frac{1}{2} w \sum ∣ d_{w} ∣ = A \subseteq W max ∣ w \in A \sum d_{w} ∣ .$
unlabeled · line 1188
$δ_{rec} = A \subseteq W max ρ sup ∣ Tr (ρ D_{A})∣ .$

Theorem: Event operator-norm formula

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