# How are measurements on $Z$ and $X$ axes interpreted in the Bloch sphere?

I'm having trouble understanding how the measurement on $$z$$ and $$x$$ axes can be interpreted in terms of the Bloch sphere representation.

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I know that the state can be written as $$∣𝜓⟩=\cos(𝜃/2)|0⟩+\exp(i𝜙)\sin(𝜃/2)|1⟩,$$ and that to make a measurement we use the squared modulus of the inner product of the basis vector and the state vector. For example, we have $$|⟨0|𝜓⟩|^2$$ for a $$z$$ measurement.

Thinking in terms of the Bloch sphere, it is clear that a rotation by an angle $$𝜙$$ would not affect a $$z$$ measurement, and this is easy to prove mathematically. But this should also be the case for a $$𝜃$$ rotation on a $$x$$ measurement, right?

But when I try to prove it mathematically, I get

$$|⟨+|𝜓⟩|^2 = \frac{1}{\sqrt2}|(\cos(𝜃/2)+\exp(i𝜙)\sin(𝜃/2)|^2 \\= \frac{1}{\sqrt2}|(\cos(𝜃/2)+\cos(𝜙)\sin(𝜃/2)+i \sin(𝜙)\sin(𝜃/2)|^2,$$

which then can be written in the form of $$a^2 + b^2$$ where $$a=\cos(𝜃/2)+\cos(𝜙)\sin(𝜃/2)$$ and $$b = \sin(𝜙)\sin(𝜃/2)$$.

This means that $$𝜃$$ affects the probability of a $$x$$ measurement. I know I must be doing something wrong but what is it?

I also don't understand why x gate is a rotation around the x axis and not the y axis ?

The $$\theta$$ and the $$\phi$$ angles are not equivalent in the Bloch sphere. First, they have different ranges -- one is $$\pi$$ and the other is $$2\pi$$. More importantly, $$\phi$$ is a rotation around a fixed axis, $$z$$, while $$\theta$$ is a rotation around a non-fixed axis that is moving with $$\phi$$. For $$\phi=0$$ this axis is $$y$$, for $$\phi=\pi/2$$ it is $$x$$, and for every other $$\phi$$ it is everything in between in the $$x-y$$ plane.
If pictorially we associate the probability of outcome to depend on the angle of the state's vector with the axis of measurement (the $$z$$ axis is the 0,1 measurement for example), then rotations around the $$z$$ axis should not affect the probabilities of 0,1 outcomes, and that's why the $$\phi$$ angle does not change this measurement. We cannot say the same about rotations by $$\theta$$ since it depends on the axis of rotation. In special cases though, for example if you set $$\phi=\pi/2$$, then the axis of $$\theta$$ rotation is $$x$$, then you should expect the +,- measurements along the $$x$$ axis not to depend on the $$\theta$$ (set $$\phi=\pi/2$$ and with basic trigo your equations should be telling you that).