Einstein De-Hass effect shows that the angular momentum associated with spin is a real angular momentum of the same nature appearing in classical mechanics (i.e rotation).
Measuring the spin state of a state superposition between spin up and spin down can yield any of them. In addition if you measured the spin in a different axis (e.g. an orthogonal axis) of such a state you can get a non-zero value. So in what since is angular momentum conserved in QM ??
The answer for this question may be in decoherence. The pure state is an ideal case. The real state is entangled with the environment and gradually becomes a totally mixed state. The angular momentum of a totally mixed state, whose density matrix is diagonal, is conserved in the classical sense.
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