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If we're measuring in common bases like $|0\rangle$, $|1\rangle$ or $|+\rangle$, $|-\rangle$ we express this by saying we're measuring with $\sigma_z$ or $\sigma_x$, or measuring in the computational or sign bases. What's the conventional or most-concise way to say we're measuring in a non-standard basis, like the in CHSH experiment where Bob measures in the computational basis rotated $\pm\frac{\pi}{8}$ radians around the y-axis? Do we derive the observable from its eigenvectors and use that?

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If you express it as an operator of the form $\vec{n}\cdot\vec{\sigma}$, it will certainly be understood. In this context, you're probably talking about $(Z+X)/\sqrt{2}$.

You could derive this from the eigenvectors, of you can derive it from the Bloch sphere picture, where a measurement corresponds to any point on the surface of the Bloch sphere (and that is specified by the vector $\vec{n}$). You just have to remember that there's a doubling of angles between the way we write them on states, and the angles on the Bloch sphere, so your $\pi/8$ angle becomes $\pi/4$.

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  • $\begingroup$ That particular axis is the rotation axis of the Hadamard gate, so you could call it the H basis. $\endgroup$ – Craig Gidney Nov 22 '18 at 18:06

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