Outcomes of quantum measurements

Question

I'm pretty new to quantum computing, and I'm wondering how I can compute the outcome of a projective measurement of a spin along the +Z axis followed by a projective measurement along the -Z axis. I think that when we make the +Z projective measurement. I know that +Z axis corresponds to $(1, 0)^{T}$ and -Z axis corresponds to the state $(0, 1)^{T}$.

I think that the answer will be $100\%$ probability of spin down and $0\%$ probability of spin up, but I'm not entirely sure why. Can someone please explain to me? I have some vague understanding that it has to do with collapsing states.

How about if we do three projective measurements: +Z followed by +X followed by -Z?

I'm not completely sure how to approach the second problem, and I'd really appreciate any help.

Answer 1

Let $\Pi_{+z}=\vert+\rangle\langle +\vert$ and $\Pi_{-z} =\vert -\rangle\langle -\vert$. It seems for any state $\vert\psi\rangle$ you want to compute \begin{align} \Pi_{-z}\Pi_{+z}\vert\psi\rangle \end{align} but this contains one piece of the form $\langle -\vert +\rangle=0$ so if $\Pi_{-z}$ immediately follows $\Pi_{+z}$ the result is $0$ so $\vert -\rangle$ has probability $0$. Alternatively, $\Pi_{+z}\vert\psi\rangle$ will project to $\vert +\rangle$ so another $\Pi_{+z}$ would give you $\vert +\rangle$ 100% of the time since the outcome of the first $\Pi_{+z}$ is $\vert +\rangle$.

You can repeat the same game with $\Pi_{+x}$ to get $\Pi_-\Pi_{+x}\Pi_+\vert\psi\rangle$ and just compute overlaps of the type $\langle +x\vert \pm z\rangle$.