Manuscript VIII-A@3 · VIII-A-v3.tex · lemma · 2398a747

Lemma: Restriction of a linear matter map

Completeness and observation invariance › Arbitrary spectator extensions · lines 976–997 · proof included or linked

VIII-A@3#A:lem:restriction, VIII-A@3#A:eq:restriction

For any linear map $Ψ : L (H_{Γ}) \to L (H_{Γ^{'}})$ and any operator $X$ on $Γ N$ ,

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

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

Proof. Pair both sides with any output operator $B$ . Both pairings are

Tr ((Ψ^{*} (B) \otimes I_{N}) X) .

Equality against all $B$ proves the identity.

Displayed formulas in this passage

VIII-A@3#A:eq:restriction · line 981
$Tr_{N} [(Ψ \otimes id_{N}) (X)] = Ψ (Tr_{N} X) .$
unlabeled · line 993
$Tr ((Ψ^{*} (B) \otimes I_{N}) X) .$