Manuscript Hawk02-B@2 · Hawk02-B.tex · proposition · 95ea744f

Proposition: Persistent weak coupling

Access controls and exact operational effects › Permanently incomplete release · lines 1341–1356 · proof included or linked

Hawk02-B@2#prop:weak

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If one cell obeys

j = 1 \sum^{\infty} sin^{2} θ_{j} \leq b < 1,

then every prefix has $q_{N} \geq 1 - b$ . Any full $K$ -cell code containing that cell has ideal recovery error at least $c_{d} (1 - b)$ .

Proof. For $x_{j} \in [0, 1]$ , $\prod_{j} (1 - x_{j}) \geq 1 - \sum_{j} x_{j}$ on each finite prefix. Use $x_{j} = sin^{2} θ_{j}$ and then (eq:manyexact).

Displayed formulas in this passage

unlabeled · line 1344
$j = 1 \sum^{\infty} sin^{2} θ_{j} \leq b < 1,$

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section: Permanently incomplete release