Manuscript X-A-ST@1 · X-A-ST.tex · proposition · d9ccbde1

Proposition: Fixed-interface error bound

A finite Enter interface for supplied evaluations › Fixed-table approximation · lines 2016–2045 · proof included or linked

X-A-ST@1#prop:table, X-A-ST@1#eq:tableerror

The map $I_{v}$ is CPTP. On the same complete interfaces,

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

Proof. The branches have Kraus operators

\frac{1 + y v _{k i}}{2} ∣ k, i, y ⟩ (⟨ τ_{k} ∣ ⟨ i ∣) \otimes I_{S} .

They are complete. Orthogonality of the output blocks bounds the trace norm of the difference, including a reference, by

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. A basis input at an index attaining the maximum saturates the bound.

Displayed formulas in this passage

X-A-ST@1#eq:tableerror · line 2020
$d_{⋄} (I_{v}, I_{v}) = \frac{1}{2} k i max ∣ v_{k i} - v_{k i} ∣, d_{⋄} (Λ, Γ) = \frac{1}{2} ∥Λ - Γ ∥_{⋄} .$
unlabeled · line 2031
$\frac{1 + y v _{k i}}{2} ∣ k, i, y ⟩ (⟨ τ_{k} ∣ ⟨ i ∣) \otimes I_{S} .$
unlabeled · line 2038
$k i max ∣ v_{k i} - v_{k i} ∣ k i \sum ∥ X_{k i} ∥_{1} .$