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Past exam of the introduction to quantum information course of the University of Oxford / 2021 / 1 / b / Solution by

Codex  0 2026-10-25

The six Pauli eigenstates have Bloch vectors

\pm e_{x}

\pm e_{y}

\pm e_{z}

X : Y : Z : ∣ \pm ⟩ = (∣0 ⟩ \pm ∣1 ⟩) / 2, ∣ \pm i ⟩ = (∣0 ⟩ \pm i ∣1 ⟩) / 2, ∣0 ⟩, ∣1 ⟩ .

(1)

Classical convex combinations of density operators produce the corresponding convex combinations of their Bloch vectors. Their convex hull is the stabilizer octahedron:

∣ s_{x} ∣ + ∣ s_{y} ∣ + ∣ s_{z} ∣ \leq 1.

(2)

The six vertices are pure Pauli states; the faces and interior represent their mixtures. To verify the reverse inclusion, give the vertex in the sign of each coordinate weight

∣ s_{j} ∣

and use the leftover weight at the origin, itself the equal mixture of either opposite pair.

Figure 1.
The standard-basis mixture diameter and the stabilizer octahedron inside the Bloch ball
. The red diameter shows all mixtures of $∣0 ⟩$ and $∣1 ⟩$ . The six Pauli vertices generate the blue octahedron. The gold point is the T-gate resource state: it lies on the Bloch sphere but outside this octahedron. **Details**.

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