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

Entropy and the Hawking-pair assumption audit

Entropy and the Hawking-pair assumption audit · lines 1076–1102

Hawk02-B@2#thm:B7, Hawk02-B@2#eq:manyentropy

For a single-cell code state $ρ_{C}$ ,

S (R) = h_{2} (q_{N}) + (1 - q_{N}) S (ρ_{C}), S (M) = h_{2} (q_{N}) + q_{N} S (ρ_{C}) .

For a pure unreferenced input, $S (R) = h_{2} (q_{N})$ . For a reference-entangled pure input, radiation need not become pure at discharge.

For identical $K$ -cell mechanisms and arbitrary reduced code state $ρ_{C^{(K)}}$ ,

S (R^{(K)}) = K h_{2} (q_{N}) + S \subseteq {1, \dots, K} \sum (1 - q_{N})^{∣ S ∣} q_{N}^{K - ∣ S ∣} S (ρ_{C_{S}}),

where the empty subsystem has entropy zero. For independent pure cell states this reduces to $K h_{2} (q_{N})$ .

Displayed formulas in this passage

Hawk02-B@2#eq:manyentropy · line 1091
$S (R^{(K)}) = K h_{2} (q_{N}) + S \subseteq {1, \dots, K} \sum (1 - q_{N})^{∣ S ∣} q_{N}^{K - ∣ S ∣} S (ρ_{C_{S}}),$
unlabeled · line 1081
$S (R) = h_{2} (q_{N}) + (1 - q_{N}) S (ρ_{C}), S (M) = h_{2} (q_{N}) + q_{N} S (ρ_{C}) .$