Manuscript En05-A@1 · En05-A.tex · lemma · f076e703

Lemma: One-layer support

Partitioned dynamics and the exact comparison theorem › Factorization and adjoint support · lines 462–483 · proof included or linked

En05-A@1#lem:onelayer

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For finite $Y$ , put

U_{s} (Y) = ⋃ {b \in P_{s} : b \cap Y \neq = \emptyset} .

Then

Φ_{s}^{*} (A_{s + 1} (Y)) \subseteq A_{s} (U_{s} (Y)) .

The reduced channel on $Y$ is determined by the channels of the blocks meeting $Y$ .

Proof. The complement of $U_{s} (Y)$ is a union of blocks. An observable on $Y$ is the identity on their output interfaces. Since the adjoint of a trace-preserving channel is unital, each complementary block adjoint contributes an identity on its input interface. The remaining adjoints act only in $U_{s} (Y)$ . Their composition determines the reduced output channel.

Displayed formulas in this passage

unlabeled · line 464
$U_{s} (Y) = ⋃ {b \in P_{s} : b \cap Y \neq = \emptyset} .$
unlabeled · line 469
$Φ_{s}^{*} (A_{s + 1} (Y)) \subseteq A_{s} (U_{s} (Y)) .$

Lemma: One-layer support

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