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If I have an arbitrary two-qubit state (in this example given from spins), and I want to measure the state of the first spin along a direction $\vec{n}$. This vector n has an angle theta with respect to the $z$-axis.

Now I want to know the possible measurement results and their probabilities, for a given theta. Unfortunately, I am not sure how to start with this, as I would usually use Born‘s rule to get the probability of measuring a certain defined state from the existing superposition.

But here, I don’t know, what the state that would be in the bra side of the Born’s rule is. Therefore, I would be really grateful for any hints on how to get from measuring along a certain direction to calculating the probability. (If necessary, I can use a certain value for theta)

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First, you need to construct the measurement projection operators. For measuring spin along $\vec{n}$, the two projection operators will be given by

$$ P_{\pm} = \frac{I \pm \vec{n}\cdot\vec{\sigma}}{2}\,.$$

Since you are only measuring the first qubit, your projection measurement operators $\{M_k\}$ will be $$\{M_k\} \equiv \{ P_+\otimes I, P_- \otimes I \}\,.$$

So, if your initial two-qubit state is $|\psi\rangle$, then you can have two outcomes, $+1$ and $-1$ and the probabilities will be given by

$$p(+1) = \langle\psi|(P_+\otimes I)|\psi\rangle\,,$$ $$p(-1) = \langle\psi|(P_-\otimes I)|\psi\rangle\,.$$

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