Bloch vector

Every qubit state has the unique formIt is pure exactly when $|r|=1$, and
$r_i=2⟨S_i⟩/\hslash$.

The Bloch sphere is the unit sphere of Bloch vectors, corresponding to pure qubit states through $\rho =(I+\mathbf{r}\cdot \mathbf{\sigma })/2$. A local unitary change of qubit basis induces a rotation of this sphere, with its common phase physically irrelevant. Mixed states occupy the interior of the unit ball.

Three Hermitian expectation values determine an arbitrary qubit state exactly when the traceless parts of the three observables span the three-dimensional space generated by the Pauli matrices. Mere linear independence as Hermitian matrices is insufficient: $I,{\sigma }_x,{\sigma }_z$ are independent but cannot detect the sign or magnitude of the Bloch $y$ component.