Manuscript Hawk04-D@1 · Hawk04-D-v1.tex · theorem · 39b16f67

Theorem: Persistent near-Bell pairs constrain entropy turnover

Stage 4: information flow, entropy tests, and Page-time limits › A joint-pair consistency test · lines 1710–1746 · proof included or linked

Hawk04-D@1#thm:pair, Hawk04-D@1#eq:pair-growth, Hawk04-D@1#eq:pair-trace

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Let $R$ be earlier radiation and $b, c$ a new outgoing qubit and partner. If

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

then

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

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

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

If successive steps have no reinteraction of earlier radiation and each right-hand side is at least $γ > 0$ , then $S (R_{N}) \geq S (R_{0}) + N γ$ .

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

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

For the trace-distance specialization, finite-dimensional entropy continuity gives $S (b c) \leq h_{2} (e) + e lo g 3$ , and partial-trace contraction gives $S (c) \geq lo g 2 - h_{2} (e)$ Audenaert. The bounds are monotone on the stated range. The final assertion is telescoping under the no-reinteraction premise.

Displayed formulas in this passage

Hawk04-D@1#eq:pair-growth · line 1718
$S (R b) - S (R) \geq lo g 2 - s - u .$
Hawk04-D@1#eq:pair-trace · line 1724
$S (R b) - S (R) \geq lo g 2 - 2 h_{2} (e) - e lo g 3.$
unlabeled · line 1714
$S (b c) \leq s, S (c) \geq lo g 2 - u,$
unlabeled · line 1737
$S (R b) + S (b c) \geq S (R) + S (c) .$

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section: A joint-pair consistency test