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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.

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  • $\begingroup$ There's an important piece of information missing: what is the initial spin state before the first measurement? $\endgroup$ – probably_someone Sep 21 '20 at 4:07
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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 $\langle +x\vert \pm z\rangle$.

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  • $\begingroup$ Ok. I see why negative z direction has probability $0$ now. I'm still trying to get the second follow-up. I'm having trouble with three measurements. $\endgroup$ – user146017 Sep 21 '20 at 5:02
  • $\begingroup$ I found that if I make a measurement along the z axis followed by x axis, then the electron will be in the up state with probability 0.5 and down state with probability 0.5. How can I extend from this result to the last measurement? $\endgroup$ – user146017 Sep 21 '20 at 5:09
  • $\begingroup$ This is not a homework site and I gave you enough to proceed. Best of luck. $\endgroup$ – ZeroTheHero Sep 21 '20 at 5:14

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