Spin projections on different axis

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I am currently trying to understand the idea of quantum entanglement (more specifically the Bell inequality). But its brought up a lack of my knowledge of spin.

So, I am trying to figure out how spin can be projected onto an arbitrary axis.

Suppose there are two entangled electrons of state 1/sqrt(2)(|up>|down> + |down>|up>). Where the first ket denotes Alice's particle and the second denotes Bobs. Now, if Alice measures spin along the z axis (where the z axis is the basis of the spins), she finds the electron in spin down state.

Now, Bob's electron must be in the spin up state. But he measures along an axis at an angle theta to the z-axis. I am having trouble picturing here how his subsequent measurement could be |up> or |down>. Since unless the measurement is being made at 90 degrees to z, there would surely only be a component of spin in the up or down direction on Bobs axis (not both).

Can anyone help me understand this? I'm finding myself incredibly confused.
 

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DrClaude
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Now, Bob's electron must be in the spin up state. But he measures along an axis at an angle theta to the z-axis. I am having trouble picturing here how his subsequent measurement could be |up> or |down>.
It is up or down with respect with the axis he is measuring at, not with respect to the z axis, to which |up> and |down> correspond.
 

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