Manuscript Hawk01-B@1 · Hawk01-B.tex · proposition · cfa976f1

Proposition: Pair-entropy obstruction

Small corrections: the changed assumption · lines 1395–1422 · proof included or linked

Hawk01-B@1#prop:pairentropy, Hawk01-B@1#eq:pairhypotheses, Hawk01-B@1#eq:pairgrowth

Let $R$ be previously emitted radiation, $b$ the next outgoing system, and $c$ its declared inaccessible partner. Suppose the actual post-step state satisfies

H (b c) \leq η, H (c) \geq h - η .

Then

H (R b) - H (R) \geq h - 2 η .

In particular, if $h > 2 η$ , radiation entropy cannot decrease at that step.

Proof. Weak monotonicity, equivalent to strong subadditivity after purification, gives

H (R b) + H (b c) \geq H (R) + H (c) .

Insert (eq:pairhypotheses).

Displayed formulas in this passage

Hawk01-B@1#eq:pairhypotheses · line 1400
$H (b c) \leq η, H (c) \geq h - η .$
Hawk01-B@1#eq:pairgrowth · line 1407
$H (R b) - H (R) \geq h - 2 η .$
unlabeled · line 1418
$H (R b) + H (b c) \geq H (R) + H (c) .$

Proposition: Pair-entropy obstruction

Displayed formulas in this passage

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