Manuscript VII-G@1 · VII-G.tex · lemma · 54f561e8

Lemma: Restriction of a selective map

Entangled spectators and complete controller blindness · lines 788–806 · proof included or linked

VII-G@1#lem:partial-trace, VII-G@1#eq:selective-restriction

For every linear map $Ψ$ on the original matter,

Tr_{N} [(Ψ \otimes id_{N}) (X)] = Ψ (Tr_{N} X) .

No trace-preservation or product-state assumption is needed.

Proof. Pair both sides with an arbitrary output operator $B$ . Their pairings equal

Tr ((Ψ^{*} (B) \otimes I_{N}) X) .

Equality against all $B$ proves the statement.

Displayed formulas in this passage

VII-G@1#eq:selective-restriction · line 791
$Tr_{N} [(Ψ \otimes id_{N}) (X)] = Ψ (Tr_{N} X) .$
unlabeled · line 802
$Tr ((Ψ^{*} (B) \otimes I_{N}) X) .$