Manuscript Hawk02-E@1 · Hawk02-E.tex · lemma · 4841592b

Lemma: Isometry-to-channel conversion

Reference-preserving finite approximation › Frequency discretization with exact packet identification · lines 1047–1065 · proof included or linked

Hawk02-E@1#lem:isometry

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If $V_{1}, V_{2} : C \to R \otimes B$ are isometries into a common 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} ∥ .

The same bound holds for the complete output channels.

Proof. For a pure code-reference input, the two output vectors differ in norm by at most $∥ V_{1} - V_{2} ∥$ . Their pure-state trace distance is no greater than that norm. Convexity handles mixed inputs, and partial trace is contractive. Taking the reference-input supremum proves the result.

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unlabeled · line 1051
$d_{⋄} (N_{V_{1}}, N_{V_{2}}) \leq ∥ V_{1} - V_{2} ∥, d_{⋄} (C_{V_{1}}, C_{V_{2}}) \leq ∥ V_{1} - V_{2} ∥ .$

Lemma: Isometry-to-channel conversion

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