Manuscript Hawk07-B@1 · Hawk07-B.tex · lemma · b9785538

Lemma: Late-radiation purity and rank

Entropy predictions and a genuinely scaled turnover theorem · lines 1107–1134 · proof included or linked

Hawk07-B@1#lem:entropy, Hawk07-B@1#eq:purity

At a cut $D = r b$ ,

E_{U} Tr [(ρ_{P}^{U})^{2}] = p_{q} (r, b, h) := \frac{r ( b ^{2} - 1 ) + b ( r ^{2} - 1 ) / h}{D ^{2} - 1} .

Moreover,

S (P) \leq lo g_{2} min (r, bh), E_{U} S (P) \geq - lo g_{2} p_{q} .

Proof. Twirl $S_{P P^{'}} \otimes I_{B B^{'}}$ . Its identity and full-swap coefficients are $r (b^{2} - 1) / (D^{2} - 1)$ and $b (r^{2} - 1) / (D^{2} - 1)$ . The vault purity is $1/ h$ , giving (eq:purity). The global $P B E$ state is pure, so the $P$ marginal has rank at most $min (r, bh)$ . Finally $S (ρ) \geq - lo g_{2} Tr ρ^{2}$ ; Jensen's inequality gives the mean lower bound.

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Hawk07-B@1#eq:purity · line 1110
$E_{U} Tr [(ρ_{P}^{U})^{2}] = p_{q} (r, b, h) := \frac{r ( b ^{2} - 1 ) + b ( r ^{2} - 1 ) / h}{D ^{2} - 1} .$
unlabeled · line 1119
$S (P) \leq lo g_{2} min (r, bh), E_{U} S (P) \geq - lo g_{2} p_{q} .$

Lemma: Late-radiation purity and rank

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