Manuscript En10-A@1 · En10-A.tex · lemma · 0a1d7743

Lemma: Complete-channel telescoping

Implementation tolerance on the same interface › Gate errors, repeated records, and exact causality · lines 1267–1287 · proof included or linked

En10-A@1#lem:telescoping

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For compatible CPTP maps $G_{1}, \dots, G_{m}$ and implemented maps $G_{1}, \dots, G_{m}$ on matching interfaces,

d_{⋄} (G_{m} \dots G_{1}, G_{m} \dots G_{1}) \leq min {1, j = 1 \sum^{m} d_{⋄} (G_{j}, G_{j})} .

Proof. Subtract the two compositions and replace one factor at a time. Each term contains one difference of corresponding maps, composed before and after with channels. Diamond norm is contractive under those channel compositions. The triangle inequality gives the sum, and half-diamond distance between channels is at most one.

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unlabeled · line 1271
$d_{⋄} (G_{m} \dots G_{1}, G_{m} \dots G_{1}) \leq min {1, j = 1 \sum^{m} d_{⋄} (G_{j}, G_{j})} .$

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section: Gate errors, repeated records, and exact causality