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

Lemma: Isometry comparison

Channels, complete residuals, and reference-stable estimates · lines 267–292 · proof included or linked

Hawk02-D@1#lem:isometry

Let $V_{1}, V_{2} : C \to R \otimes B$ be isometries into the same output space. Then

d_{⋄} (N_{V_{1}}, N_{V_{2}}) \leq ∥ V_{1} - V_{2} ∥, d_{⋄} (C_{V_{1}}, C_{V_{2}}) \leq ∥ V_{1} - V_{2} ∥ .

Proof. For any unit vector $ψ$ on the code and a reference, put $v_{j} = (V_{j} \otimes I) ψ$ . The trace distance of their pure-state density operators is

1 - ∣ ⟨ v_{1}, v_{2} ⟩ ∣^{2} \leq ∥ v_{1} - v_{2} ∥ \leq ∥ V_{1} - V_{2} ∥ .

Convexity handles mixed inputs. Partial trace is contractive for trace distance. Taking the supremum over reference-entangled inputs proves both inequalities.

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unlabeled · line 271
$d_{⋄} (N_{V_{1}}, N_{V_{2}}) \leq ∥ V_{1} - V_{2} ∥, d_{⋄} (C_{V_{1}}, C_{V_{2}}) \leq ∥ V_{1} - V_{2} ∥ .$
unlabeled · line 284
$1 - ∣ ⟨ v_{1}, v_{2} ⟩ ∣^{2} \leq ∥ v_{1} - v_{2} ∥ \leq ∥ V_{1} - V_{2} ∥ .$

Lemma: Isometry comparison

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