Ramana Kumar comments on Eight Definitions of Observability

Ramana Kumar 29 Jan 2021 10:52 UTC
LW: 3 AF: 2
AF
And it seems we do actually need ${A s s u m e}_{S_{1}} (C) & {A s s u m e}_{S_{2}} (C) ≃ {A s s u m e}_{S_{1} \cup S_{2}} (C)$ in the proof to justify:
Thus it suffices to show that $(D_{1} & 1_{R_{2} \cup R_{3}}) \otimes (D_{2} & 1_{R_{1} \cup R_{3}}) ≃ D_{1} & D_{2} & 1_{R_{3}}$ .
Without it, we have to show $(D_{1} & 1_{R_{2} \cup R_{3}}) \otimes (D_{2} & 1_{R_{1} \cup R_{3}}) ≃ {A s s u m e}_{S_{1} \cup S_{2}} (C) & 1_{R_{3}}$ instead.
- Ramana Kumar 30 Jan 2021 0:12 UTC
  LW: 3 AF: 2
  AF Parent
  UPDATE: I was able to prove ${A s s u m e}_{S_{1}} (C) & {A s s u m e}_{S_{2}} (C) ≃ {A s s u m e}_{S_{1} \cup S_{2}} (C)$ in general whenever $S_{1}$ and $S_{2}$ are disjoint and both in $O b s (C)$ , with help from Rohin Shah, following the “restrict attention to world $S_{1} \cup S_{2}$ ” approach I hinted at earlier.