Manuscript X-B-ST@1 · X-B-ST.tex · proposition · d03c150c

Proposition: Finite-table channel and error

Installed evaluations and a complete clock instrument › A fixed-interface approximation theorem · lines 2403–2437 · proof included or linked

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

Back to manuscript

The map $I_{v}$ 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 identity holds with arbitrary retained references.

Proof. Use Kraus operators

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

Their completeness is immediate. For an input operator $X$ , with a reference included, the difference has orthogonal output 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 trace-norm contraction of pinching, which is an average of unitary conjugations. A basis input at an index attaining the maximum saturates the bound. Divide by two.

Displayed formulas in this passage

X-B-ST@1#eq:tableerror · line 2407
$d_{⋄} (I_{v}, I_{v}) = \frac{1}{2} k max ∣ v_{k} - v_{k} ∣, d_{⋄} (Λ, Γ) = \frac{1}{2} ∥Λ - Γ ∥_{⋄} .$
unlabeled · line 2419
$p_{y} (v_{k}) ∣ k, y ⟩_{R} \otimes D_{k} .$
unlabeled · line 2425
$\frac{y ( v _{k} - v _{k} )}{2} X_{k k} .$
unlabeled · line 2429
$k max ∣ v_{k} - v_{k} ∣ k \sum ∥ X_{k k} ∥_{1} \leq k max ∣ v_{k} - v_{k} ∣ ∥ X ∥_{1} .$

Proposition: Finite-table channel and error

Displayed formulas in this passage

Nearby in this section