Manuscript XXII-D@1 · XXII-D.tex · lemma · 17889d18

Lemma: Full comparison implies logical and referenced comparison

Complete-channel transport and error assembly › Reference marginal versus logical transport · lines 794–819 · proof included or linked

XXII-D@1#lem:full-logical, XXII-D@1#eq:reference-error

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d_{⋄} (Γ_{n}, Θ_{n}) \leq e_{n}, Γ_{n} : L (M) \to L (M \otimes F_{n}),

then the logical reduction $Λ_{n} = Tr_{F_{n}} Γ_{n}$ satisfies

d_{⋄} (Λ_{n}, id_{M}) \leq e_{n} .

For every state $ρ_{M R}$ ,

d_{tr} ((Λ_{n} \otimes id_{R}) (ρ_{M R}), ρ_{M R}) \leq e_{n} .

Proof. Partial trace is a channel, so postcomposition contracts the diamond norm. Tensoring with the reference and evaluating on a state of trace norm one gives (eq:reference-error).

Displayed formulas in this passage

unlabeled · line 797
$d_{⋄} (Γ_{n}, Θ_{n}) \leq e_{n}, Γ_{n} : L (M) \to L (M \otimes F_{n}),$
unlabeled · line 804
$d_{⋄} (Λ_{n}, id_{M}) \leq e_{n} .$
XXII-D@1#eq:reference-error · line 808
$d_{tr} ((Λ_{n} \otimes id_{R}) (ρ_{M R}), ρ_{M R}) \leq e_{n} .$

Lemma: Full comparison implies logical and referenced comparison

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