Manuscript En10-B@1 · En10-B.tex · lemma · a1105bad

Lemma: Finite comparison ledger

Complete-channel tolerance and layer separation › Compatible interfaces are required for telescoping · lines 1360–1380 · proof included or linked

En10-B@1#lem:telescoping

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For compatible CPTP factors $G_{1}, \dots, G_{s}$ and implemented factors $G_{1}, \dots, G_{s}$ on identified matching interfaces,

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

Proof. Subtract the two compositions and replace one factor at a time. Every term contains one factor difference, before and after which are channels. Diamond norm is contractive under those compositions. The triangle inequality gives the sum. Half-diamond distance between channels is at most one.

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unlabeled · line 1365
$d_{⋄} (G_{s} \dots G_{1}, G_{s} \dots G_{1}) \leq min {1, j = 1 \sum^{s} d_{⋄} (G_{j}, G_{j})} .$

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section: Compatible interfaces are required for telescoping