Manuscript VII-J-ST@1 · VII-J-ST.tex · definition · f1f1d82c

Definition: Reference-assisted signalling

Complete channels and intervention-based influence · lines 439–460

VII-J-ST@1#def:signal, VII-J-ST@1#eq:signal

For input region $A$ , output region $B$ , and $Φ_{B} = Tr_{B^{c}} Φ$ , define

Sig_{Φ} (A \to B) = E, ρ, I_{A}, J_{A} sup d_{tr} ((Φ_{B} \otimes id_{E}) (ρ_{I}), (Φ_{B} \otimes id_{E}) (ρ_{J})),

where $E$ is any finite untouched reference, $ρ$ is any input-reference state, and

ρ_{I} = (I_{A} \otimes id_{A^{c} E}) (ρ) .

The interventions $I_{A}, J_{A}$ are deterministic CPTP maps on the same interface. They do not act on $E$ .

Displayed formulas in this passage

VII-J-ST@1#eq:signal · line 443
$Sig_{Φ} (A \to B) = E, ρ, I_{A}, J_{A} sup d_{tr} ((Φ_{B} \otimes id_{E}) (ρ_{I}), (Φ_{B} \otimes id_{E}) (ρ_{J})),$
unlabeled · line 454
$ρ_{I} = (I_{A} \otimes id_{A^{c} E}) (ρ) .$