Manuscript Hawk04-C@1 · Hawk04-C-v1.tex · lemma · 94d46352

Lemma: Ignored local operations do not signal across a fixed cut

Row 2: the event-horizon analogy · lines 675–705 · proof included or linked

Hawk04-C@1#lem:local, Hawk04-C@1#eq:no-signalling

Let $Λ : L (B) \to L (B^{'})$ be CPTP. For every state on $Q \otimes R \otimes B$ ,

Tr_{B^{'}} ((id_{QR} \otimes Λ) (ρ)) = Tr_{B} ρ .

Proof. If ${L_{j}}$ is a Kraus family of $Λ$ , then $\sum_{j} L_{j}^{†} L_{j} = I_{B}$ . For every observable $A$ on $Q \otimes R$ ,

Tr [(A \otimes I_{B^{'}}) (id_{QR} \otimes Λ) (ρ)] = j \sum Tr [(A \otimes L_{j}^{†} L_{j}) ρ] = Tr [(A \otimes I_{B}) ρ] .

All expectations on $QR$ agree, proving (eq:no-signalling).

Displayed formulas in this passage

Hawk04-C@1#eq:no-signalling · line 679
$Tr_{B^{'}} ((id_{QR} \otimes Λ) (ρ)) = Tr_{B} ρ .$
unlabeled · line 691
$Tr [(A \otimes I_{B^{'}}) (id_{QR} \otimes Λ) (ρ)] = j \sum Tr [(A \otimes L_{j}^{†} L_{j}) ρ] = Tr [(A \otimes I_{B}) ρ] .$