Manuscript Oai365-A@1 · Oai365-A.tex · lemma · a3840e3d

Lemma: Complete-channel telescoping

Finite-shot acquisition and complete-channel error › Complete instruments rather than scalar records · lines 1226–1244 · proof included or linked

Oai365-A@1#lem:telescoping

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If corresponding stages have half-diamond allowances $η_{1}, \dots, η_{L}$ , then

d_{⋄} (Φ_{L} \dots Φ_{1}, Φ_{L} \dots Φ_{1}) \leq min {1, η_{1} + \dots + η_{L}} .

The allowance bounds all output event-probability differences and all untouched-reference output trace distances.

Proof. Telescope the difference of compositions. CPTP pre- and post-composition contract the diamond norm. The diamond norm includes arbitrary untouched references, and measurement cannot increase trace distance.

Displayed formulas in this passage

unlabeled · line 1230
$d_{⋄} (Φ_{L} \dots Φ_{1}, Φ_{L} \dots Φ_{1}) \leq min {1, η_{1} + \dots + η_{L}} .$

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section: Complete instruments rather than scalar records
section: Complete instruments rather than scalar records