The Exercise

The Pauli matrices $X$, $Y$ and $Z$ each have two normalized eigenvectors, one with eigenvalue $+1$ and one with eigenvalue $-1$: six states of a qubit in all. The exercise asks for the points of the Bloch sphere that represent them. The library states, for each of the six, its eigenvalue and its Bloch vector.

A Qubit and Its States

A quantum system $S$ is a physical system whose state space is a finite dimensional complex Hilbert space.^✦ The qubit is the two level system whose state space is $\mathbb{C}^2$ with its standard inner product.^✦

A pure state $\lvert\psi\rangle$ of $S$ is a vector of unit norm in the state space of $S$.^✦ A state $\rho$ of $S$ is a positive operator of unit trace on the state space of $S$.^✦ A pure state $\lvert\psi\rangle$ of $S$ determines the state $\lvert\psi\rangle\langle\psi\rvert$ of $S$, the rank one projector onto $\lvert\psi\rangle$.^✦

The computational basis states $\lvert 0\rangle$ and $\lvert 1\rangle$ of the qubit are the pure states whose vectors are the standard basis vectors $(1,0)$ and $(0,1)$ of $\mathbb{C}^2$. The other four states of the exercise are the pure states of the qubit with vectors $$\lvert\pm\rangle = \frac{1}{\sqrt{2}}\bigl(\lvert 0\rangle \pm \lvert 1\rangle\bigr), \qquad \lvert\pm i\rangle = \frac{1}{\sqrt{2}}\bigl(\lvert 0\rangle \pm i\lvert 1\rangle\bigr).$$(1)

The Pauli Observables

An observable $A$ on $S$ is a self adjoint operator on the state space of $S$, $A^\dagger = A$.^✦ A pure state $\lvert\psi\rangle$ of $S$ is an eigenstate of an observable $A$ on $S$ with eigenvalue $\mu$, a real number, when $$A\lvert\psi\rangle = \mu\lvert\psi\rangle.$$(2)^✦

The Pauli matrices are the complex $2\times 2$ matrices $$X = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}, \qquad Y = \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix}, \qquad Z = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix},$$(3) and $\vec{\sigma} = (X, Y, Z)$ is the Pauli vector. For a real vector $\vec{n} = (n_1, n_2, n_3)$, the matrix $\vec{n}\cdot\vec{\sigma}$ is the complex $2\times 2$ matrix $$\vec{n}\cdot\vec{\sigma} = n_1 X + n_2 Y + n_3 Z.$$(4)^✦

The Pauli observable $\vec{n}\cdot\vec{\sigma}$ is the observable on the qubit whose operator is the matrix (4) acting on $\mathbb{C}^2$.^✦ The observables $X$, $Y$ and $Z$ are the Pauli observables for the unit vectors $\hat{n} = (1,0,0)$, $(0,1,0)$ and $(0,0,1)$, with the matrices (3) as their operators.

The Bloch Sphere

For a real vector $\vec{r} = (r_1, r_2, r_3)$ in the closed unit ball, $r_1^2 + r_2^2 + r_3^2 \le 1$, the Bloch state $\rho(\vec{r})$ is the state of the qubit whose operator is the Bloch matrix $$\rho(\vec{r}) = \frac{1}{2}\bigl(I + \vec{r}\cdot\vec{\sigma}\bigr),$$(5)^✦ with $I$ the $2\times 2$ identity. The vector $\vec{r}$ is the Bloch vector of the state, and the unit sphere $\lvert\vec{r}\rvert = 1$ is the Bloch sphere.

The Claim

What Is Checked

The claim is stated over the qubit, its six pure states $\lvert 0\rangle$, $\lvert 1\rangle$, $\lvert\pm\rangle$ and $\lvert\pm i\rangle$ with the states $\lvert\psi\rangle\langle\psi\rvert$ they determine, the Pauli observables $X$, $Y$ and $Z$ with the eigenstate condition (2), and the Bloch states (5) at the six unit vectors along the coordinate axes.

About these constructions the library proves that each of the six vectors has unit norm, and that the projector onto a unit vector is a state, a positive operator of unit trace. The matrix $\vec{n}\cdot\vec{\sigma}$ is Hermitian, so each Pauli observable is self adjoint. The Bloch matrix has trace one and is positive exactly when $\lvert\vec{r}\rvert \le 1$, which makes $\rho(\vec{r})$ a state for every $\vec{r}$ in the closed unit ball.