Consider the qubit $|\psi\rangle$ in it's canonical form: $$ |\psi\rangle = \cos(\theta/2)|0\rangle+e^{i\phi}\sin(\theta/2)|1\rangle $$ associated with the point $\big(\sin(\theta)\cos(\phi),\sin(\theta)\sin(\phi),\cos(\theta)\big)$ in Bloch's sphere. What are the points associated with the qubits: $$ X|\psi\rangle\text{ , }Y|\psi\rangle\text{ , }Z|\psi\rangle $$ where X, Y and Z are pauli matrices.
How am I able to do that? I know how Pauli matrices change the basis qubits... For example, $X$ applications changes $|0\rangle \longrightarrow |1\rangle$ and $|1\rangle \longrightarrow |0\rangle$. But after applying X, Y and Z, I just don't know how to interpret its coefficients as coordinates in Bloch sphere.
Can someone please help me with that?
Thanks!
For example, we compute $$ X |\psi\rangle = \pmatrix{0&1\\1&0}[\cos(\theta/2)|0\rangle+e^{i\phi}\sin(\theta/2)|1\rangle]\\ = \cos(\theta/2)|1\rangle+e^{i\phi}\sin(\theta/2)|0\rangle \\ = e^{i\phi}\sin(\theta/2)|0\rangle + \cos(\theta/2)|1\rangle $$ However, this state is not in the required canonical form since the coefficient of $|0\rangle$ has a non-zero complex phase. So, we multiply the entire vector by an appropriate complex phase (which is to say we divide the whole vector by $e^{i\phi}$) to get $$ \sin(\theta/2)|0\rangle + e^{-i\phi}\cos(\theta/2)|1\rangle $$ Now, in order to convert this vector into Bloch-coordinates, we need to write this in the canonical form of $$ \cos(\hat \theta/2) |0\rangle + e^{i\hat \phi} \sin(\hat \theta/2) | 1 \rangle $$ for angles $\hat \theta, \hat \phi$. We associate this with the point $\big(\sin(\hat \theta)\cos(\hat\phi),\sin(\hat \theta)\sin(\hat \phi),\cos(\hat \theta)\big)$. To that end, we note that $$ \sin (\theta/2) = \cos((\pi - \theta)/2),\\ \cos (\theta/2) = \sin((\pi - \theta)/2),\\ e^{-i\phi} = e^{i(2\pi - \phi)} $$ And with this association, we can write $$ e^{-i\phi} X | \psi \rangle = \cos(\hat \theta/2) |0\rangle + e^{i\hat \phi} \sin(\hat \theta/2) | 1 \rangle $$ where $\hat \theta = \pi - \theta$ and $\hat\phi = (2 \pi - \phi)$.