Manuscript VII-E@1 · VII-E.tex · lemma · b713cf3c

Lemma: One-layer support

Prescribed kinematics and schedule dependence › Exact backwards support · lines 656–671 · proof included or linked

VII-E@1#lem:one-layer

For a block channel $Φ = ⨂_{B \in B} Φ_{B}$ ,

Φ^{*} (A (Y)) \subseteq A B \cap Y \neq = \emptyset ⋃ B .

Proof. The complement of the displayed union is a union of blocks on which the input observable is the identity. Each corresponding adjoint maps the identity to itself. The other blocks can enlarge support only within their own union.

Displayed formulas in this passage

unlabeled · line 659
$Φ^{*} (A (Y)) \subseteq A B \cap Y \neq = \emptyset ⋃ B .$

Nearby in this section

section: Exact backwards support
theorem: Theorem: Prescribed schedule locality
corollary: Corollary: Readout dependence
section: Exact backwards support