The Exercise

The pure states of a qubit are pictured as the points of a sphere, each at a polar angle $\theta$ and an azimuth $\phi$. The exercise extends the picture to mixed states. It asks to show that every density matrix of a qubit can be written in terms of a real vector $\vec{r}$ in three dimensions, its Bloch vector, of length at most one, and that the state is pure exactly when that length is one. The zero vector should give the state $I/2$, and a pure state pictured at $\theta$ and $\phi$ should have as Bloch vector the point of the sphere at the same angles.

The library states the four parts as theorems about the states of a qubit.

A Qubit

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

A state $\rho$ of $S$ is a positive operator of unit trace on the state space of $S$.^✦ A pure state $|\psi\rangle$ of $S$ is a unit vector in the state space of $S$; it determines the state $|\psi\rangle\langle\psi|$ of $S$, the projector onto that vector. A state $\rho$ of $S$ is pure when $\rho = |\psi\rangle\langle\psi|$ for some pure state $|\psi\rangle$ of $S$.^✦ The maximally mixed state of $S$ is the state $I/d$, with $d$ the dimension of the state space of $S$; for the qubit it is $I/2$.^✦

Bloch States

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) and $\vec{\sigma} = (X, Y, Z)$ is the Pauli vector. For a real vector $\vec{r} = (r_x, r_y, r_z)$ in three dimensions, the matrix $\vec{r} \cdot \vec{\sigma}$ is $r_x X + r_y Y + r_z Z$, and the Bloch matrix of $\vec{r}$ is $$\frac{1}{2}\bigl(I + \vec{r} \cdot \vec{\sigma}\bigr).$$(2) For a real vector $\vec{r}$ of length at most one, $\lVert \vec{r} \rVert^2 = r_x^2 + r_y^2 + r_z^2 \le 1$, the Bloch state $\rho_{\vec{r}}$ is the state of the qubit whose matrix in the basis $|0\rangle$, $|1\rangle$ is the Bloch matrix (2).^✦

For real angles $\theta$ and $\phi$, the pure state $|\psi(\theta,\phi)\rangle$ of the qubit is the unit vector $$|\psi(\theta,\phi)\rangle = \cos\frac{\theta}{2}\,|0\rangle + e^{i\phi}\sin\frac{\theta}{2}\,|1\rangle,$$(3) and the unit vector $\hat{r}(\theta,\phi)$ in three dimensions at polar angle $\theta$ and azimuth $\phi$ is $$\hat{r}(\theta,\phi) = (\sin\theta\cos\phi,\ \sin\theta\sin\phi,\ \cos\theta).$$(4)

The Claim

What Is Checked

The claim is stated over the states of the qubit, the Bloch states $\rho_{\vec{r}}$ built from the Pauli matrices, the pure states $|\psi(\theta,\phi)\rangle$ with their projectors and the vectors $\hat{r}(\theta,\phi)$, and the notions of a pure state and of the maximally mixed state.

The library proves that the Bloch matrix of a vector of length at most one is positive with unit trace, so that $\rho_{\vec{r}}$ is a state of the qubit, and that the projector onto a unit vector and the operator $I/d$ are density operators as well. Both (3) and (4) are proved to have length one.