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To make matter simple, I just assume it to have zero spin. So let's say I did and the measured spin is zero.It depends, did you measure the spin of the electron in the initial ground state?

What puzzles me is how the spin information carried by the electron can be transferred to the emitted photon given that the photon is emitted in a random direction. For example, if the emitted photon has spin +1, can we conclude that the absorbed photon must have spin +1, regardless of the angle between the two paths?Information is not lost by absorption, it is just carried by another carrier. Information carried by a photon before absorption is carried by electron after the absorption.