Manuscript En05-C@1 · En05-C.tex · lemma · aa213ea6

Lemma: Localized internal-device support

Stage E: one-layer locality without product channels · lines 1311–1340 · proof included or linked

En05-C@1#lem:devicesupport, En05-C@1#eq:enlargement, En05-C@1#eq:exteriorfix

Let $A$ be an isotone locally commuting net. For the device (eq:internaldevice),

T_{b} (A (Y)) \subseteq A (Y \cup b) .

If $b \cap Y = \emptyset$ , then

T_{b} ∣_{A (Y)} = id .

Devices with disjoint localization regions commute as global superoperators.

Proof. For $a \in A (Y)$ , isotony places $a$ and all $V_{b, μ}$ in $A (Y \cup b)$ . This algebra is closed under products and sums, proving (eq:enlargement).

For disjoint $b, Y$ , local commutativity gives

T_{b} (a) = a μ \sum V_{b, μ}^{*} V_{b, μ} = a .

For disjoint devices $b, c$ , their Kraus operators and adjoints commute. Expanding both compositions and reordering those operators proves $T_{b} T_{c} = T_{c} T_{b}$ .

Displayed formulas in this passage

En05-C@1#eq:enlargement · line 1315
$T_{b} (A (Y)) \subseteq A (Y \cup b) .$
En05-C@1#eq:exteriorfix · line 1320
$T_{b} ∣_{A (Y)} = id .$
unlabeled · line 1334
$T_{b} (a) = a μ \sum V_{b, μ}^{*} V_{b, μ} = a .$

Lemma: Localized internal-device support

Displayed formulas in this passage

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