Manuscript IX-D-ST@1 · IX-D-ST.tex · proposition · b1dc631a

Proposition: Fixed-interface approximation

Shared finite Enter certificate interface › A fixed finite-table comparison · lines 2317–2345 · proof included or linked

IX-D-ST@1#prop:table

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The channel $I_{v}$ is CPTP. With the same labels, pinching, report function, and payload interface,

d_{⋄} (I_{v}, I_{v}) = \frac{1}{2} k i max ∣ v_{k i} - v_{k i} ∣, d_{⋄} (Φ, Ψ) = \frac{1}{2} ∥Φ - Ψ ∥_{⋄} .

Proof. Kraus operators are

p_{y} (k, i) ∣ k, i, y, r (k, i)⟩ (⟨ τ_{k} ∣ ⟨ i ∣) \otimes I_{P} .

They are complete. Including a reference, orthogonal output blocks make the trace norm of the difference at most

k i max ∣ v_{k i} - v_{k i} ∣ k i \sum ∥ X_{k i} ∥_{1} .

Pinching is an average of unitary conjugations and contracts trace norm even on arbitrary operators. A basis input at a maximizing index attains the bound.

Displayed formulas in this passage

unlabeled · line 2322
$d_{⋄} (I_{v}, I_{v}) = \frac{1}{2} k i max ∣ v_{k i} - v_{k i} ∣, d_{⋄} (Φ, Ψ) = \frac{1}{2} ∥Φ - Ψ ∥_{⋄} .$
unlabeled · line 2331
$p_{y} (k, i) ∣ k, i, y, r (k, i)⟩ (⟨ τ_{k} ∣ ⟨ i ∣) \otimes I_{P} .$
unlabeled · line 2338
$k i max ∣ v_{k i} - v_{k i} ∣ k i \sum ∥ X_{k i} ∥_{1} .$

Proposition: Fixed-interface approximation

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