  Multiple systems hosting Andreev molecular states have been proposed and
studied, consisting of closely spaced Josephson junctions modeled as ballistic
channels. We show that replacing the ballistic channels in the weak link of the
Josephson junctions with quantum dots (QD), leads to a very exciting, rich
phase diagram. It shows a strong nonlocal Josephson effect: as one junction is
tuned the current-phase relation of the other junction is modified. This
architecture hosts $0 - \pi$ transitions and shows a tunable anomalous
phase-shift $\phi_0$ , nonlocally controlled in both cases, without relying on
spin-orbit interaction or Zeeman fields. In addition significant
superconducting diode effect can also be observed. The presented non-local
current-phase relation can be used as a signature of the formation of an
Andreev molecular state, as well as to introduce new ways to tune quantum
architectures.
