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

Proposition: Discarded radiation

Access controls and exact operational effects › Incomplete collection · lines 1197–1219 · proof included or linked

Hawk02-B@2#prop:collection

Collect only cells in $S \subseteq {1, \dots, N}$ . Set

p_{S} = j \in S \sum p_{j}, q_{S} = 1 - p_{S} .

The collected channel is an isometric erasure channel with

ε_{rec} (N_{S}) = q_{S} c_{d} .

The complete residual includes memory and uncollected radiation.

Proof. In (eq:Wunnormalized), all collected emission amplitudes share a vacuum residual. The remaining amplitudes share collected vacuum and have residual support orthogonal to that vacuum. Tracing the residual gives the erasure-channel form with success weight $p_{S}$ . Apply Theorem thm:B3. If $p_{S} = 0$ , the channel is constant.

Displayed formulas in this passage

unlabeled · line 1201
$p_{S} = j \in S \sum p_{j}, q_{S} = 1 - p_{S} .$
unlabeled · line 1206
$ε_{rec} (N_{S}) = q_{S} c_{d} .$