# Convention for expressing measurement in non-standard basis

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?

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 $$\pi/4$$.

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