Projection on an axis

Question

Can we say that the coordinates in 3-space corresponding to the Bloch sphere, by taking expectation values of the Pauli spin operators is the same as the projections onto the different axis? \begin{align} x&=\langle \psi | \sigma_x |\psi \rangle \\ y&=\langle \psi | \sigma_y |\psi \rangle \\ z&=\langle \psi | \sigma_z |\psi \rangle \end{align}

Answer 1

Yes, the identities you give are all correct. This also extends to the case where the state is mixed (instead of pure), in which case the density matrix can be written, through $$ \rho=\begin{pmatrix}\rho_{00} & \rho_{01}\\ \rho_{01}^* & \rho_{11}\end{pmatrix} =\frac12 \mathbb I+ \vec r\cdot\vec\sigma, $$ as a linear combination of the identity and the Pauli matrices, where $\vec r$ is the position of the state on the Bloch sphere (on the surface for pure states, and in the interior for mixed states) with components given by $$ x_i=\mathrm{Tr}\mathopen{}\left(\rho\sigma_i\right)\mathclose{}. $$