Manuscript Hawk05-C@1 · Hawk05-C.tex · theorem · ba0719f1

Theorem: Two-curve extrapolation budget

Typicality and physical transfer require different allowances › Finite-dimensional entropy continuity · lines 849–885 · proof included or linked

Hawk05-C@1#thm:transfer, Hawk05-C@1#eq:Qbudget, Hawk05-C@1#eq:Ybudget

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Under Definition def:comparison, for the declared fixed pure input,

∣ E S (R_{phys}) - min (A, B) ∣ ∣ E H (Y_{phys}) - A ∣ \leq c_{L} (η) + τ lo g_{2} m + u (r, b) + max (e_{R}, e_{B}), \leq c_{K} (min {1, η + ν}) + τ lo g_{2} r + \frac{1}{b ln 2} + e_{R} .

The record-only recovery obstruction remains independent of these entropy allowances.

Proof. The channel comparison bounds the compared output trace distances, including references, by $η$ . Detector composition contraction and the triangle inequality give record distance at most $min {1, η + ν}$ . Apply the finite entropy continuity bounds and take expectations. Proposition prop:TV replaces the proxy ensemble mean by its Haar mean. Proposition prop:deficits compares those means with $lo g_{2} m$ and $lo g_{2} r$ . Finally, $min$ is one-Lipschitz in the maximum norm:

∣ min (lo g_{2} r, lo g_{2} b) - min (A, B) ∣ \leq max (e_{R}, e_{B}) .

This proves both budgets. The classical recovery lower bound is Proposition prop:classical.

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Hawk05-C@1#eq:Qbudget · line 853
$∣ E S (R_{phys}) - min (A, B) ∣ ∣ E H (Y_{phys}) - A ∣ \leq c_{L} (η) + τ lo g_{2} m + u (r, b) + max (e_{R}, e_{B}), \leq c_{K} (min {1, η + ν}) + τ lo g_{2} r + \frac{1}{b ln 2} + e_{R} .$
Hawk05-C@1#eq:Ybudget · line 853
$∣ E S (R_{phys}) - min (A, B) ∣ ∣ E H (Y_{phys}) - A ∣ \leq c_{L} (η) + τ lo g_{2} m + u (r, b) + max (e_{R}, e_{B}), \leq c_{K} (min {1, η + ν}) + τ lo g_{2} r + \frac{1}{b ln 2} + e_{R} .$
unlabeled · line 878
$∣ min (lo g_{2} r, lo g_{2} b) - min (A, B) ∣ \leq max (e_{R}, e_{B}) .$

Theorem: Two-curve extrapolation budget

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