Spin passing through wire loop

Consider a spin-1/2 particle passing through a wire loop with a light bulb in series:

The spin produces a magnetic moment and therefore a magnetic field, which can induce current in the loop due to changing flux. For a pure upwards-pointing spin $\lvert\uparrow\rangle$ (shown above), we have an increasing upwards flux before it enters and a decreasing flux afterwards, so the light bulb turns on before and after entering the loop. For $\lvert\downarrow\rangle$ the flux is reversed but the light bulb still turns on at the same times.

Now let the spin part of the wavefunction be $\frac{1}{\sqrt{2}}(\lvert\uparrow\rangle + \lvert\downarrow\rangle)$. Does the light bulb turn on?