Manuscript Oai283-A@1 · Oai283-A.tex · lemma · 6fb75895

Lemma: Channel telescoping

Composition, workspace, and implementation allowances › Complete-channel telescoping · lines 469–484 · proof included or linked

Oai283-A@1#lem:telescoping

For compatible finite sequences of channels,

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

Proof. Insert one implemented stage at a time. The difference telescopes, and CPTP pre- and post-composition contract the diamond norm.

Displayed formulas in this passage

unlabeled · line 472
$d_{⋄} (Φ_{L} \dots Φ_{1}, Φ_{L} \dots Φ_{1}) \leq min {1, j = 1 \sum^{L} d_{⋄} (Φ_{j}, Φ_{j})} .$