The Exercise

A qubit is rotated twice, first by an angle $\beta_1$ about a unit axis $\hat n_1$ , then by an angle $\beta_2$ about a unit axis $\hat n_2$ . The two rotations together make a single rotation $R_{\hat n_{12}}(\beta_{12})$ by an angle $\beta_{12}$ about an axis $\hat n_{12}$ , and the exercise finds it from the two it is made of: the cosine $c_{12} = \cos(\beta_{12}/2)$ of its half angle and the vector $s_{12}\hat n_{12}$ , with $s_{12} = \sin(\beta_{12}/2)$ , are given in terms of the half angles and the axes of the two rotations. A second part takes equal angles, $\beta_1 = \beta_2 = \beta$ , with the first axis along the $z$ axis, the unit vector $\hat z = (0, 0, 1)$ .

In the library the first part is an identity between $2 \times 2$ matrices, which expresses the product of the two rotations through $c_{12}$ and $s_{12}\hat n_{12}$ , both defined from the two angles and axes. The second part becomes a statement about those two definitions, and an example shows that one term of $s_{12}\hat n_{12}$ cannot carry the sign opposite to the one in the library.

Rotations of a Qubit

The Pauli matrices $X$ , $Y$ and $Z$ are the complex $2 \times 2$ 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) and $\vec\sigma = (X, Y, Z)$ is the Pauli vector. For a real vector $\vec n = (n_x, n_y, n_z)$ in three dimensions, the matrix $\vec n \cdot \vec\sigma$ is the complex $2 \times 2$ matrix $n_x X + n_y Y + n_z Z$ .^✦

The rotation operator $R_{\vec n}(\theta)$ about an axis $\vec n$ , a real vector in three dimensions, by a real angle $\theta$ is the complex $2 \times 2$ matrix given by the matrix exponential $R_{\vec n}(\theta) = \exp\!\left(-\frac{i\theta}{2}\, \vec n \cdot \vec\sigma\right),$ (2)^✦ written $R_{\hat n}(\theta)$ when the axis is a unit vector $\hat n$ .

For real angles $\beta_1, \beta_2$ and real vectors $\vec n_1, \vec n_2$ in three dimensions, write $c_i = \cos(\beta_i/2)$ and $s_i = \sin(\beta_i/2)$ . The composed cosine $c_{12}$ of the pairs $(\beta_1, \vec n_1)$ and $(\beta_2, \vec n_2)$ is the real number $c_{12} = c_1 c_2 - s_1 s_2\, \vec n_1 \cdot \vec n_2 ,$ (3) and their composed axis vector $s_{12}\hat n_{12}$ is the real vector in three dimensions $s_{12}\hat n_{12} = s_1 c_2\, \vec n_1 + c_1 s_2\, \vec n_2 + s_1 s_2\, \vec n_2 \times \vec n_1 .$ (4)

The Claim

Theorem 1. Let $\hat n_1$ and $\hat n_2$ be unit vectors and $\beta_1$ , $\beta_2$ real angles. Then $R_{\hat n_2}(\beta_2)\, R_{\hat n_1}(\beta_1) = c_{12}\, I - i\, (s_{12}\hat n_{12}) \cdot \vec\sigma ,$ (5)^✦ where $I$ is the $2 \times 2$ identity and $c_{12}$ and $s_{12}\hat n_{12}$ are the composed cosine (3) and the composed axis vector (4) of $(\beta_1, \hat n_1)$ and $(\beta_2, \hat n_2)$ .

The cross product is the one term of (4) that changes sign when the two pairs trade places. The book’s equation (4.20) prints it with a minus sign, $-s_1 s_2\, \hat n_2 \times \hat n_1$ .

Theorem 2. The formula with the printed sign fails for the product in the order of (5): for $\hat n_1 = \hat z = (0, 0, 1)$ , $\hat n_2 = \hat x = (1, 0, 0)$ and $\beta_1 = \beta_2 = \pi$ , $R_{\hat x}(\pi)\, R_{\hat z}(\pi) \neq c_{12}\, I - i \left( s_1 c_2\, \hat z + c_1 s_2\, \hat x - s_1 s_2\, \hat x \times \hat z \right) \cdot \vec\sigma ,$ (6)^✦ where $c_1 = c_2 = \cos(\pi/2)$ , $s_1 = s_2 = \sin(\pi/2)$ and $c_{12}$ is the composed cosine of $(\pi, \hat z)$ and $(\pi, \hat x)$ .

Theorem 3. Let $\beta$ be a real angle and $\vec n_2$ a real vector in three dimensions, and write $c = \cos(\beta/2)$ and $s = \sin(\beta/2)$ . The composed cosine and the composed axis vector of $(\beta, \hat z)$ and $(\beta, \vec n_2)$ are $c_{12} = c^2 - s^2\, \hat z \cdot \vec n_2 ,$ (7)^✦ $s_{12}\hat n_{12} = s c\, (\hat z + \vec n_2) + s^2\, \vec n_2 \times \hat z .$ (8)^✦

What Is Checked

The claim is stated over the rotation operators $R_{\hat n}(\theta)$ of (2), built from the matrices $\hat n \cdot \vec\sigma$ and the Pauli matrices (1), and over the composed cosine (3) and the composed axis vector (4).

About these constructions the library proves the composition law (5) and the counterexample (6) to the printed sign. For equal angles with the first axis along $\hat z$ it proves the reduced forms (7) and (8).