The Exercise

Two unit vectors $\hat m$ and $\hat n$ in three dimensions fix two axes of rotation on the Bloch sphere. The question is whether every unitary $U$ on one qubit can be built from rotations about these two axes alone: a phase, then a rotation about $\hat n$ , a rotation about $\hat m$ and a rotation about $\hat n$ again, with four real angles chosen for $U$ . The library states that perpendicular axes always suffice, and that axes which are merely not parallel need not.

Rotations About an Axis

The Pauli matrices $X$ , $Y$ and $Z$ are the $2 \times 2$ complex matrices $X = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}, \qquad Y = \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix}, \qquad Z = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}.$ (1) The matrix $\vec n \cdot \vec\sigma$ of a real vector $\vec n = (n_0, n_1, n_2)$ in three dimensions is the $2 \times 2$ complex matrix $\vec n \cdot \vec\sigma = n_0 X + n_1 Y + n_2 Z.$ (2)^✦ The rotation operator $R_{\vec n}(\theta)$ about the axis $\vec n$ by a real angle $\theta$ is the $2 \times 2$ complex matrix $R_{\vec n}(\theta) = \exp\!\left(-\frac{i\theta}{2}\, \vec n \cdot \vec\sigma\right);$ (3)^✦ for a unit vector $\hat n$ it is written $R_{\hat n}(\theta)$ .

The Claims

Theorem 1. Let $\hat m$ and $\hat n$ be perpendicular unit vectors in three dimensions, $\hat m \cdot \hat n = 0$ , and let $U$ be a unitary $2 \times 2$ complex matrix. Then there are real angles $\alpha$ , $\beta$ , $\gamma$ , $\delta$ with $U = e^{i\alpha}\, R_{\hat n}(\beta)\, R_{\hat m}(\gamma)\, R_{\hat n}(\delta).$ (4)^✦

Theorem 2. There are unit vectors $\hat m$ and $\hat n$ in three dimensions that are not parallel, $\hat m \neq c\, \hat n$ for every real $c$ , and a unitary $2 \times 2$ complex matrix $U$ such that no real angles $\alpha$ , $\beta$ , $\gamma$ , $\delta$ satisfy $U = e^{i\alpha}\, R_{\hat n}(\beta)\, R_{\hat m}(\gamma)\, R_{\hat n}(\delta).$ (5)^✦

What Is Checked

Each is stated over the rotation operator (3), and through it over the matrix $\vec n \cdot \vec\sigma$ of (2) and the Pauli matrices (1), with $U$ ranging over the unitary $2 \times 2$ complex matrices.