Manuscript Hawk02-D@1 · Hawk02-D.tex · proposition · e6e4b72f

Proposition: Residual omission counterexample

Debugging the proposed imports and physical interpretation › Source marginal relaxation is not complete decoupling · lines 1654–1673 · proof included or linked

Hawk02-D@1#prop:omission

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A constant source marginal does not imply small complete residual leakage.

Proof. Let

V ∣ ψ ⟩ = ∣ 0 ⟩_{S} ∣ 0 ⟩_{R} ∣ ψ ⟩_{E} .

The source marginal is the fixed state $∣ 0 ⟩ ⟨ 0 ∣_{S}$ . The accessible radiation is also fixed. The complete residual contains $E$ and retains the entire code. By Lemma lem:replacement, its distance from any replacement channel is at least $1 - d^{- 2}$ , attained by the maximally mixed replacement on $E$ .

Displayed formulas in this passage

unlabeled · line 1662
$V ∣ ψ ⟩ = ∣ 0 ⟩_{S} ∣ 0 ⟩_{R} ∣ ψ ⟩_{E} .$

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section: Source marginal relaxation is not complete decoupling