The Exercise

A unit vector $\hat{n}$ in three dimensions and the vector $\vec{\sigma} = (X, Y, Z)$ of Pauli matrices give the qubit observable $\hat{n} \cdot \vec{\sigma}$, the spin along $\hat{n}$. For a function $f$ from the complex numbers to the complex numbers and a real number $\theta$, the exercise is to show that the matrix $f(\theta\, \hat{n} \cdot \vec{\sigma})$ is a combination of the identity $I$ and $\hat{n} \cdot \vec{\sigma}$ whose two coefficients depend on $f$ only through its values $f(\theta)$ and $f(-\theta)$.

Functions of a Hermitian Matrix

For a Hermitian complex $m \times m$ matrix $A$ and a function $f$ from the complex numbers to the complex numbers, the function $f(A)$ of $A$ is the $m \times m$ matrix that has the eigenvectors of $A$, with the eigenvalue $f(\lambda)$ where $A$ has the eigenvalue $\lambda$.^✦ Written with the real eigenvalues $\lambda_1, \dots, \lambda_m$ of $A$ and a unitary $U$ whose columns are orthonormal eigenvectors for them, it is $$f(A) = U \operatorname{diag}\bigl(f(\lambda_1), \dots, f(\lambda_m)\bigr)\, U^\dagger .$$(1)^✦

The Pauli Matrices

The Pauli matrices are the complex $2 \times 2$ matrices $$X = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}, \qquad Y = \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix}, \qquad Z = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}.$$(2) For a real vector $\vec{n} = (n_1, n_2, n_3)$ in three dimensions, the matrix $\vec{n} \cdot \vec{\sigma}$ is the complex $2 \times 2$ matrix whose coefficients on the Pauli matrices are the components of $\vec{n}$: $$\vec{n} \cdot \vec{\sigma} = n_1 X + n_2 Y + n_3 Z .$$(3)^✦

The Claim

What Is Checked

The claim is stated over the function of a Hermitian matrix (1), taken at $A = \theta\, \hat{n} \cdot \vec{\sigma}$, and over the matrix $\vec{n} \cdot \vec{\sigma}$ of (3), built from the Pauli matrices (2).

About these constructions the library proves that $\theta\, \vec{n} \cdot \vec{\sigma}$ is Hermitian for every real $\theta$ and every real vector $\vec{n}$, of unit length or not, so that the left side of (4) is defined.^✦