Bloch ball ($\{\mathbf{s}\in {\mathbb{R}}^3:|\mathbf{s}|\le 1\}$)

The qubit density operators $\rho =(I+\mathbf{s}\cdot \mathbf{\sigma })/2$ correspond exactly to the closed unit ball: their eigenvalues are $(1\pm |\mathbf{s}|)/2$, so positivity is equivalent to $|\mathbf{s}|\le 1$. The boundary is the Bloch sphere of pure states, and the centre is $I/2$. A depolarizing channel scales this ball isotropically, possibly with a sign reversal in the larger completely positive parameter interval.

The convex hull of the six Pauli eigenstates in the Bloch ball. A mixture has a Bloch vector obeying the displayed inequality. Conversely, assign weights $|s_x|,|s_y|,|s_z|$ to the corresponding signed vertices and distribute the leftover weight over any opposite pair; this realizes every vector satisfying the inequality. Single-qubit Clifford gates permute its vertices, while a T gate can take a vertex to a pure state outside it.

For a qubit density matrix $\rho =(I+n\cdot \sigma )/2$ with nonzero Bloch vector, $\hat{n}=n/\Vert n\Vert$ and the multiplication rule for the Pauli matrices give these orthogonal rank-one spectral projectors, with eigenvalues $(1\pm \Vert n\Vert )/2$. Their Bloch vectors are opposite unit vectors $\pm \hat{n}$. At $n=0$ the state is maximally mixed and the direction is undefined; any orthonormal eigenbasis gives an antipodal pair, but no pair is distinguished by the state.