Manuscript X-F-ST@1 · X-F-ST.tex · proposition · e71f03ef

Proposition: Reference-stable report comparison

Finite supplied evaluations and the complete Enter interface › An explicit report encoding and its error · lines 2331–2356 · proof included or linked

X-F-ST@1#prop:reports

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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 ∥_{⋄} .

Proof. Use the complete Kraus family $p_{y} (v_{j}) ∣ j, y ⟩_{R} \otimes D_{j}$ . For an arbitrary input operator 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 inequality is pinching contraction. A clock-basis input at a maximizing index saturates the estimate.

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unlabeled · line 2334
$d_{⋄} (I_{v}, I_{v}) = \frac{1}{2} j max ∣ v_{j} - v_{j} ∣, d_{⋄} (Λ, Γ) = \frac{1}{2} ∥Λ - Γ ∥_{⋄} .$
unlabeled · line 2349
$j max ∣ v_{j} - v_{j} ∣ j \sum ∥ X_{j j} ∥_{1} \leq j max ∣ v_{j} - v_{j} ∣ ∥ X ∥_{1} .$

Proposition: Reference-stable report comparison

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