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

Reference marginal versus logical transport

Complete-channel transport and error assembly › Reference marginal versus logical transport · lines 758–793

XXII-D@1#eq:reference-marginal, XXII-D@1#eq:identity, XXII-D@1#eq:complete-target

For every channel $Λ : L (M) \to L (M^{'})$ ,

Tr_{M^{'}} [(Λ \otimes id_{R}) (ρ_{M R})] = ρ_{R} .

This follows from trace preservation. It does not establish preservation of the message–reference correlations.

After a declared identification $M^{'} ≅ M$ , identity transport is the stronger channel equality

Λ = id_{M}, (Λ \otimes id_{R}) (X) = X for every X .

A replacement channel satisfies (eq:reference-marginal) and generally destroys those correlations. This is the distinction of XXII-B's Candidate D CheungXXIIB and the reference-stable transport statements of En08-B CheungEn08B (lem:identity).

For a declared residual and record system $F_{n}$ , a complete ideal transport target is

Θ_{n} (X) = X \otimes σ_{n},

where $σ_{n}$ is input independent. It may include a declared arrival-record law. The logical target is $id_{M}$ , obtained by tracing out $F_{n}$ .

Displayed formulas in this passage

XXII-D@1#eq:reference-marginal · line 761
$Tr_{M^{'}} [(Λ \otimes id_{R}) (ρ_{M R})] = ρ_{R} .$
XXII-D@1#eq:identity · line 771
$Λ = id_{M}, (Λ \otimes id_{R}) (X) = X for every X .$
XXII-D@1#eq:complete-target · line 786
$Θ_{n} (X) = X \otimes σ_{n},$

Reference marginal versus logical transport

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