Manuscript Hawk03-A@1 · Hawk03-A.tex · theorem · 9a109c12

Theorem: Quantitative independent-factor benchmark

Independent-factor AMPS benchmarks · lines 367–399 · proof included or linked

Hawk03-A@1#thm:amps, Hawk03-A@1#eq:ampsbound, Hawk03-A@1#eq:ampsnecessary

Let $∣ β ⟩_{A B}$ be a pure state, let

s = S (B)_{β} > 0,

and suppose

d_{tr} (ρ_{A B}, ∣ β ⟩ ⟨ β ∣) \leq ε .

Under the independent-factor hypothesis,

S (B ∣ E)_{ρ} \geq s - g_{d} (ε) .

If a proposed radiation-purification certificate also requires

S (B ∣ E)_{ρ} \leq - μ, μ \geq 0,

then its simultaneous validity requires

g_{d} (ε) \geq s + μ .

Proof. For the pure pair, $S (B ∣ A)_{β} = - s$ . Lemma lem:continuity gives $S (B ∣ A)_{ρ} \leq - s + g_{d} (ε)$ . Lemma lem:weak then gives (eq:ampsbound). Combining this lower bound with the asserted upper bound $- μ$ gives (eq:ampsnecessary).

Displayed formulas in this passage

Hawk03-A@1#eq:ampsbound · line 378
$S (B ∣ E)_{ρ} \geq s - g_{d} (ε) .$
Hawk03-A@1#eq:ampsnecessary · line 387
$g_{d} (ε) \geq s + μ .$
unlabeled · line 370
$s = S (B)_{β} > 0,$
unlabeled · line 374
$d_{tr} (ρ_{A B}, ∣ β ⟩ ⟨ β ∣) \leq ε .$
unlabeled · line 383
$S (B ∣ E)_{ρ} \leq - μ, μ \geq 0,$

Theorem: Quantitative independent-factor benchmark

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