Manuscript X-G-ST@1 · X-G-ST.tex · proposition · 3c97567f

Proposition: Complete report error

Stage 5: supplied evaluations and a complete Enter instrument › One explicit report encoding · lines 2275–2302 · proof included or linked

X-G-ST@1#prop:report

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The channel $I_{v}$ is CPTP and

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

The comparison includes all untouched references.

Proof. A complete Kraus family is $p_{y} (v_{j}) ∣ j, y ⟩_{R} \otimes D_{j}$ . On an arbitrary input with reference, the difference has orthogonal blocks $y (v_{j} - v_{j}) X_{j j} /2$ . Its trace norm is bounded by

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

The last step is pinching contraction. A basis input at a maximizing $j$ attains the bound. Divide by two.

Displayed formulas in this passage

unlabeled · line 2278
$d_{⋄} (I_{v}, I_{v}) = \frac{1}{2} j max ∣ v_{j} - v_{j} ∣, d_{⋄} (Λ, Γ) = \frac{1}{2} ∥Λ - Γ ∥_{⋄} .$
unlabeled · line 2294
$j max ∣ v_{j} - v_{j} ∣ j \sum ∥ X_{j j} ∥_{1} \leq j max ∣ v_{j} - v_{j} ∣ ∥ X ∥_{1} .$

Proposition: Complete report error

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