Let's say I have a quantum state such as $$ | \psi \rangle = \alpha |00\rangle + \beta |11\rangle $$ for some pair of qubits. I am wondering how to interpret an operator like $$ P_1 = | 0 0 \rangle \langle 0 0 | + | 1 1 \rangle \langle 1 1 | , $$ or $$ P_2 = | 1 0 \rangle \langle 1 0 | + | 0 1 \rangle \langle 0 1 | , $$ which might be applied to such a state. (States and operators similar to these occur in discussions of quantum bit-flip error correction which is why I am thinking about this).
Is it physically possible to perform the operation corresponding to, e.g., $P_1$? It looks like this means, "Use a device to measure the state of the qubits. If the device finds $|00\rangle$ or $|11\rangle$ then it must react the same way physically, giving no indication of which of these two it found. It might display a "1" (for measurement result 1) on a read-out screen in this case. If it finds one of $|01\rangle$ or $|10\rangle$ instead, it should react in some other way, displaying maybe a "2" (measurement result 2) on the screen."
One example I've thought up that might be an illustration of this is a system where the total energy depends on the relative spin of two spin-1/2 states. If they are parallel, meaning $|00\rangle$ or $|11\rangle$, then there is an energy, say $+J$. If instead, they are antiparallel, i.e. $|10\rangle$ or $|01\rangle$ then it has an energy $-J$. Then the measurement device I am trying to construct above would simply measure the total energy of the pair and read "1" if it measures $+J$ and "2" if it measures "-J".
Is this device physically plausible or is it just a fantasy?