(Local) Conservation of Energy in Quantum Mechanics 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?
 A: As for all EPR-type situations, the answer is in the local statistics over an ensemble of identical copies. Say you do measure one pair of systems and find that system 1 is state $|0\rangle$ while its counterpart across the hall, system 2, is found in state $|E\rangle$. You can say that there is some "spooky-energy-transfer-at-a-distance" by means of wave-function collapse. Great. Now try to get the next pair of systems to do exactly the same thing and cement the result. 
You'll find out that it is impossible: there is no way to predict in what state you will measure the 1st system, and this means, unfortunately, no way to predict the energy of the 2nd system. The only thing you can do is average your results over as many attempts as possible. When you do that, you simply find that the average energy both for the measured systems and for their counterparts across the hall is E/2. The only conclusion available is that "on average energy is conserved", although in individual pairs it is redistributed. 
This is no different than any attempt to use entanglement and projection-at-a-distance for faster-than-light communication. Energy transfer is subject to the same rules. 
A: This is not a problem for mainstream interpretations of quantum mechanics where there is no physical collapse of wavefunctions.
It's only a problem for fringe theories, such as spontaneous localization (aka objective collapse), where quantum mechanics is modified to induce a real collapse.  From what I understand, violating conservation of energy has always been the biggest problem with those set of interpretations (which, strictly speaking should not be considered interpretations but rather, extensions of quantum mechanics).
If the wavefunction is considered to be real, as in Everett, then it never collapses at all.  So energy and information flow is always local. 
If, on the other hand, the wavefunction is assumed instead to be epistemic, as in Copenhagen or QBism, 
then the situation is analogous to classical mechanics when you have some uncertainty about a state.  For example, if you don't know whether a coin is heads or tails, then finding out instantly from a friend who looked at it gives you information and "collapses" the 2 states you were imagining each with 50% probability into a single known state. The information flow associated with that collapse is nonlocal in a sense, but not one that's relevant to causality or physics. In your example, the expected value of energy E/2 is nothing more than your expectation that it might be 0 and it might be E.  You don't know until you make an observation.
It's only the people who try to modify quantum mechanics (usually, by adding non-linear terms to the Schrodinger Equation) to make this collapse physical who run into a conflict with locality and energy conservation. 
A: Conservation of Energy is problematic in the Quantum world. Noether's approach applied to the quantum world only gives energy conservation on average, which falls short of what we want. Further suppose |E1> and |E2> are energy eigenvectors of a system that starts off in state |E1>. Measurement are then made of a non-commuting observable quantity A followed by a second measurement of energy. In general, energy will not be conserved. The usual explanation is that the measuring device imparted/absorbed energy - the system was not closed. But it is difficult to make the conservation of energy in closed systems rigorous. I believe it can be done is specific cases. I don't think interpretations have much to do with it.
Energy is not necessarily conserved in General Relativity (GTR). If Quantum Mechanics is a more fundamental theory that GTR, and is essentially correct, then it should not be possible to "prove" conservation of energy in Quantum Mechanics for the most general case. 
