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

Proposition: Accessible-output criterion

Record compatibility, residuals, and composition › Accessible output algebras · lines 1286–1307 · proof included or linked

VII-E@1#prop:accessible, VII-E@1#eq:observable-norm

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The channels agree on $O$ for every input if and only if

Θ^{*} (A) = Ξ^{*} (A) (A \in O) .

Moreover,

d_{O} (Θ, Ξ) = \frac{1}{2} A = A^{†} \in O ∥ A ∥_{\infty} \leq 1 sup ∥ Θ^{*} (A) - Ξ^{*} (A) ∥_{\infty} .

Proof. For Hermitian $A$ , equality of all input expectations is equality of the Hermitian adjoint images. General algebra elements are complex linear combinations of Hermitian elements. For the distance, interchange the two suprema and optimize the Hermitian expectation over states.

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VII-E@1#eq:observable-norm · line 1293
$d_{O} (Θ, Ξ) = \frac{1}{2} A = A^{†} \in O ∥ A ∥_{\infty} \leq 1 sup ∥ Θ^{*} (A) - Ξ^{*} (A) ∥_{\infty} .$
unlabeled · line 1289
$Θ^{*} (A) = Ξ^{*} (A) (A \in O) .$

Proposition: Accessible-output criterion

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