Bloch vector
= Bloch vector
{c}
Every qubit state has the unique form
$$
\rho=\frac12(I+r_x\sigma_x+r_y\sigma_y+r_z\sigma_z),
\qquad |r|\leq1.
$$
It is pure exactly when $|r|=1$, and
$r_i=2\langle S_i\rangle/\hbar$.
Unit-length Bloch vectors form the <Bloch sphere>; mixed-state vectors occupy the ball’s interior.