# Teleportation and non-orthogonal measurement operators

Consider the simple state teleportation gadget below. Measuring the entangled state after the CZ gate in the $$X$$ eigenbasis (equivalent to first applying $$H$$ and then measuring in computational basis) teleports the state $$|\psi\rangle$$ to the second register up to a $$X^m H$$.

Now, what if we were to consider a measurement of non-orthogonal operators (for simplicity say a POVM given by $$E_1, E_2, E_3$$) instead of measuring in the $$X$$ basis. I am reading an article in which the authors seem to indicate that this will generally leave the state on the second register in a mixed state.

Could someone explain why this is the case?

(I have actually taken this slightly out of context, as this originally concerns a quantum optics measurement, namely a so-called heterodyne measurement, where the measurement operators are given by $$E=|\alpha\rangle\langle\alpha|$$ but the coherent states $$|\alpha\rangle$$ are not orthogonal and form an overcomplete basis. However, I believe this should boil down to the same question)