The Exercise

A state of a qubit is fixed by its Bloch vector, a real vector $\vec r$ in three dimensions of length at most one: the density operator is $\frac12(I + \vec r\cdot\vec\sigma)$, with $\vec\sigma$ the vector of Pauli matrices. A unitary $U$ acting on the qubit takes the state $\rho$ to $U\rho U^\dagger$, which has a Bloch vector of its own. The exercise asks for the map from the old Bloch vector to the new one: for each $U$ there is a single rotation $R$ of three dimensional space that carries every Bloch vector $\vec r$ to the Bloch vector $R\vec r$ of the evolved state.

A Qubit and Its States

A quantum system $S$ is described by its state space, a finite dimensional complex Hilbert space.^✦ The qubit is the two level system whose state space is $\mathbb{C}^2$ itself, with its standard basis $|0\rangle$, $|1\rangle$.^✦ A state $\rho$ of $S$ is a positive operator of unit trace on the state space of $S$.^✦

The Pauli matrices are $$X = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}, \qquad Y = \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix}, \qquad Z = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}.$$(1) For a real vector $\vec r = (r_1, r_2, r_3)$ in three dimensions, $\vec r\cdot\vec\sigma$ is the complex $2\times 2$ matrix $r_1X + r_2Y + r_3Z$, and the Bloch matrix of $\vec r$ is $\frac12(I + \vec r\cdot\vec\sigma)$: $$\tfrac12\left(I + \vec r\cdot\vec\sigma\right) = \frac12\begin{pmatrix} 1 + r_3 & r_1 - i r_2 \\ r_1 + i r_2 & 1 - r_3 \end{pmatrix}.$$(2) For a vector $\vec r$ in the closed unit ball, $|\vec r|^2 = r_1^2 + r_2^2 + r_3^2 \le 1$, the Bloch state $\rho(\vec r)$ is the state of the qubit whose operator is the Bloch matrix (2) acting on $\mathbb{C}^2$.^✦

Unitary Gates

A unitary $U$ of a quantum system $S$, a closed system evolution of $S$, is an operator $U$ on the state space of $S$ with $U^\dagger U = UU^\dagger = I$.^✦ Under $U$ a state $\rho$ of $S$ evolves as $$\rho \;\longmapsto\; U\rho\,U^\dagger.$$(3)^✦

On the qubit, a unitary $2\times 2$ complex matrix $U$, with $U^\dagger U = UU^\dagger = I$, gives the single qubit gate $U$: the unitary of the qubit whose operator is the linear map of $\mathbb{C}^2$ that the matrix represents in the standard basis, $$U|j\rangle = \sum_{k=0}^{1} U_{kj}\,|k\rangle, \qquad j = 0, 1.$$(4)^✦

The Claim

A rotation of three dimensional space is a real $3\times 3$ matrix $R$ with $R^{T}R = RR^{T} = I$ and $\det R = 1$; the rotations form the special orthogonal group $SO(3)$.

What Is Checked

The claim is stated over the single qubit gate (4) of a unitary matrix $U$, the evolution (3) of a state under a unitary, the Bloch states $\rho(\vec r)$ of the qubit and the rotations in $SO(3)$.

The library proves that the Bloch matrix of $\vec r$ has unit trace and is positive exactly when $|\vec r| \le 1$, which makes $\rho(\vec r)$ a state. Conjugation by a unitary takes a density operator to a density operator, so $U\rho U^\dagger$ is again a state, and the operator of the gate is unitary because the matrix $U$ is.