Manuscript VIII-ST@1 · VIII-ST.tex · proposition · fe979f3b

Proposition: Accessible-observable criterion

Named access, residual instruments, and recovery boundaries · lines 1073–1098 · proof included or linked

VIII-ST@1#prop:accessible

For channels $Φ, Ψ$ with common interfaces and a fixed Hermitian output observable $J$ ,

Tr (J Φ (ρ)) = Tr (J Ψ (ρ)) for every ρ ⟺ Φ^{*} (J) = Ψ^{*} (J) .

Moreover,

ρ sup ∣ Tr (J (Φ - Ψ) (ρ)) ∣ = ∥ Φ^{*} (J) - Ψ^{*} (J) ∥_{op} .

For a restricted input class, only annihilation of its real linear span follows. For the complete classical block-diagonal class, replace the operator difference by its block pinching.

Proof. The difference is Hermitian. Its largest absolute expectation is its spectral norm, attained in an extremal eigenspace. The restricted statements follow by testing only the declared span; block-diagonal states test exactly the pinched operator.

Displayed formulas in this passage

unlabeled · line 1077
$Tr (J Φ (ρ)) = Tr (J Ψ (ρ)) for every ρ ⟺ Φ^{*} (J) = Ψ^{*} (J) .$
unlabeled · line 1084
$ρ sup ∣ Tr (J (Φ - Ψ) (ρ)) ∣ = ∥ Φ^{*} (J) - Ψ^{*} (J) ∥_{op} .$

Proposition: Accessible-observable criterion

Displayed formulas in this passage

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