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I have a $|+\rangle$ state qubit and I measure it in a random basis. The random basis is made with random $\theta$, $\varphi$ and $\lambda$ of $U3$ gate. How can I calculate the theoretical probabilities for this basis.

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The theoretical probability is calculated by looking at the projection of state $|+\rangle$ on random basis.

Let the random basis be written as $$|M\rangle=e^{i\lambda}(\cos(\theta /2) |0\rangle+ e^{i\phi}\sin(\theta /2)|1\rangle)\,.$$

Then the probability of measuring state $|+\rangle$ is given by: $$|\langle M|+ \rangle|^2=1/2+\cos(\phi)\cos(\theta /2)\sin(\theta /2)\,.$$

For example, if $\theta=\pi/2,\phi=0$ with any arbitrary global phase $\lambda$, we have $$|M\rangle=|+\rangle\,,$$ and the probability would be 1.

For $\theta=\pi/2, \phi=\pi$, $$|M\rangle=|-\rangle\,,$$ and the probability would be 0.

I have assumed that the initial state of the qubit is $|0\rangle$ before the $U3$ gate application to determine the basis. If not we will have to consider initial state probabilities into account as well.

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