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

Proposition: Untouched remote reduction

Operational locality with explicit controller support › Nonselective remote invariance · lines 1450–1472 · proof included or linked

VII-G@1#prop:remote

For a local instrument ${Ψ_{a}^{R}}$ on $R$ , with $\sum_{a} Ψ_{a}^{R}$ trace preserving,

Tr_{R} [a \sum (Ψ_{a}^{R} \otimes id_{R^{c}}) (ρ)] = Tr_{R} ρ .

This holds for entangled inputs. It does not assert equality of conditional remote states.

Proof. Pair the remote output with $B$ . Its expectation is

Tr ([a \sum Ψ_{a}^{R *} (I_{R})] \otimes B ρ) = Tr ((I_{R} \otimes B) ρ) .

Displayed formulas in this passage

unlabeled · line 1454
$Tr_{R} [a \sum (Ψ_{a}^{R} \otimes id_{R^{c}}) (ρ)] = Tr_{R} ρ .$
unlabeled · line 1466
$Tr ([a \sum Ψ_{a}^{R *} (I_{R})] \otimes B ρ) = Tr ((I_{R} \otimes B) ρ) .$