Manuscript Hawk02-A@1 · Hawk02-A.tex · theorem · 7fbae745

Theorem: Pair-entropy growth interface

Stage 3: pair structure and an explicit entropy test · lines 728–772 · proof included or linked

Hawk02-A@1#thm:pairentropy, Hawk02-A@1#eq:pairgrowth, Hawk02-A@1#eq:tracepairgrowth

At an emission cut, let $R$ be the already emitted radiation and let $b, c$ be a new outgoing qubit and a partner qubit. Suppose

S (b c) \leq s, S (c) \geq lo g 2 - t,

with natural logarithms. Then

S (R b) - S (R) \geq lo g 2 - s - t .

If, additionally,

d_{tr} (ρ_{b c}, ∣ Φ^{+} ⟩ ⟨ Φ^{+} ∣) \leq e \leq \frac{1}{2},

then one may take

s = h_{2} (e) + e lo g 3, t = h_{2} (e),

where $h_{2} (e) = - e lo g e - (1 - e) lo g (1 - e)$ . Hence

S (R b) - S (R) \geq lo g 2 - 2 h_{2} (e) - e lo g 3.

Proof. Weak monotonicity, a consequence of strong subadditivity LiebRuskai, gives

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

Substitution proves (eq:pairgrowth).

For the trace-distance statement, entropy continuity in dimension four bounds $S (b c)$ by $h_{2} (e) + e lo g 3$ . Partial trace contracts trace distance, so $ρ_{c}$ is within $e$ of $I /2$ . The dimension-two entropy continuity bound gives $S (c) \geq lo g 2 - h_{2} (e)$ Audenaert. Both continuity allowances are monotone for the stated range.

Displayed formulas in this passage

Hawk02-A@1#eq:pairgrowth · line 737
$S (R b) - S (R) \geq lo g 2 - s - t .$
Hawk02-A@1#eq:tracepairgrowth · line 750
$S (R b) - S (R) \geq lo g 2 - 2 h_{2} (e) - e lo g 3.$
unlabeled · line 733
$S (b c) \leq s, S (c) \geq lo g 2 - t,$
unlabeled · line 742
$d_{tr} (ρ_{b c}, ∣ Φ^{+} ⟩ ⟨ Φ^{+} ∣) \leq e \leq \frac{1}{2},$
unlabeled · line 746
$s = h_{2} (e) + e lo g 3, t = h_{2} (e),$
unlabeled · line 761
$S (R b) + S (b c) \geq S (R) + S (c) .$

Theorem: Pair-entropy growth interface

Displayed formulas in this passage

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