Manuscript Hawk07-A@1 · Hawk07-A.tex · proposition · f9c0f5da

Proposition: No signaling by an unreported local operation

The causal overlay and its communication ledger › Entanglement does not provide free signaling · lines 1747–1764 · proof included or linked

Hawk07-A@1#prop:nosignal

For every bipartite state $ρ_{A B}$ and TP local channel $Λ_{A}$ ,

Tr_{A} [(Λ_{A} \otimes id_{B}) (ρ_{A B})] = Tr_{A} ρ_{A B} .

Thus a Bell resource alone cannot signal a later choice at $A$ to $B$ .

Proof. For every $B$ -observable $Z$ , the pulled-back observable is

Λ_{A}^{*} (I_{A}) \otimes Z = I_{A} \otimes Z .

Expectations and hence reduced states agree.

Displayed formulas in this passage

unlabeled · line 1751
$Tr_{A} [(Λ_{A} \otimes id_{B}) (ρ_{A B})] = Tr_{A} ρ_{A B} .$
unlabeled · line 1760
$Λ_{A}^{*} (I_{A}) \otimes Z = I_{A} \otimes Z .$