Manuscript X-C-ST@1 · X-C-ST.tex · proposition · b679954e

Proposition: Fixed-interface table error

Finite installed encoding and the clock instrument › One explicit finite-table error topology · lines 2311–2346 · proof included or linked

X-C-ST@1#prop:table, X-C-ST@1#eq:table-error

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The channel $I_{v}$ is CPTP and, on the same complete interfaces,

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

The equality includes arbitrary untouched references.

Proof. Use the complete Kraus family

p_{y} (v_{k}) ∣ k, y ⟩_{R} \otimes D_{k} .

With a reference included, the difference on an operator $X$ has orthogonal blocks

\frac{y ( v _{k} - v _{k} )}{2} X_{k k} .

Its trace norm is at most

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

The last inequality is pinching contraction: pinching is an average of unitary conjugations. A clock-basis input at an index attaining the maximum saturates the bound. Division by two proves the formula.

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X-C-ST@1#eq:table-error · line 2315
$d_{⋄} (I_{v}, I_{v}) = \frac{1}{2} k max ∣ v_{k} - v_{k} ∣, d_{⋄} (Λ, Γ) = \frac{1}{2} ∥Λ - Γ ∥_{⋄} .$
unlabeled · line 2328
$p_{y} (v_{k}) ∣ k, y ⟩_{R} \otimes D_{k} .$
unlabeled · line 2333
$\frac{y ( v _{k} - v _{k} )}{2} X_{k k} .$
unlabeled · line 2337
$k max ∣ v_{k} - v_{k} ∣ k \sum ∥ X_{k k} ∥_{1} \leq k max ∣ v_{k} - v_{k} ∣ ∥ X ∥_{1} .$

Proposition: Fixed-interface table error

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