Generally, we say that conservation of energy is a local law; the change in energy in some small region of space is equal to the energy flux out of that region. However, in quantum mechanics, we can have superpositions of energy states. Then, when we measure them, they "instantly" achieve a certain energy. I'm not sure how to reconcile this with local energy conservation.
To be specific, let's consider the following case: we have two identical copies of some two-state system with energy levels $0$ and $E$, and we prepare them in an entangled state given by
$$ |0E\rangle +|E0\rangle $$
Let's assume one atom is in our lab, the other is across the hall. Then locally, their (expected) energies before measurement are each $E/2$. If we measure the electron in our lab, it instantaneously has energy $0$ or $E$--and the same thing happens across the hall! It seems like if we replace "energy density" with "expected energy density", we can have discontinuous jumps in the energy.
Is there any way to formulate local energy conservation in quantum mechanics? Especially if we assume nothing has interacted with the electron across the hall?