I'm having trouble understanding how to put together two one-bit operators to get a two-bit operator. For example, suppose I have two electrons in the spin state:
If I'm understanding things correctly, then to measure the state of the first one (along the z axis) I would tensor $\sigma_z$ with the identity operator. This seems to work.
My problem comes in if I want to measure the state of both of them. I would have thought that I would tensor $\sigma_z$ with itself. But when I do that I get an operator that has degenerate eigenvalues. In other words, I get an eigenvalue of $+1$ for either of the possible outcomes. I'm not sure if the whole approach is wrong, or if I'm just making a mistake somewhere.
I hope this description is clear. I'm somewhat of a novice at this. Also I imagine that there might be a better existing tag for this question than "linear-algebra" but I couldn't find it. Any comments appreciated.