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

Proposition: Mode mismatch

Access controls and exact operational effects › Incorrect radiation mode · lines 1273–1290 · proof included or linked

Hawk02-B@2#prop:mismatch

For $p > 0$ , let $v$ be normalized and $J ∣ a ⟩ = \sum_{j} v_{j} ∣ a; j ⟩$ . The coherent-mode decoder using $J$ instead of $J_{N}$ has exact error

[1 - p ∣ ⟨ v, v ⟩ ∣^{2}] c_{d} .

At $p = 0$ its error is $c_{d}$ .

Proof. $J^{†} J_{N} = ⟨ v, v ⟩ I_{C}$ . Compression transfers the stated weight and replaces its complement by $I_{C} / d$ . Apply Lemma lem:replacement.

Displayed formulas in this passage

unlabeled · line 1279
$[1 - p ∣ ⟨ v, v ⟩ ∣^{2}] c_{d} .$