The Exercise

Exercise 2.11 is about the three Pauli matrices $X$, $Y$ and $Z$, which act on the state space $\mathbb{C}^2$ of a qubit. It asks for the eigenvalues and eigenvectors of each matrix, and for its diagonal representation: the matrix written as a sum, over its eigenvectors $|\lambda\rangle$, of the eigenvalue $\lambda$ times the outer product $|\lambda\rangle\langle\lambda|$.

The library answers with the eigenvectors of $X$ and of $Y$ written out as column vectors, the eigenvalue equation of each, and the diagonal representation of all three matrices. $Z$ is already diagonal in the computational basis, the basis of the column vectors $|0\rangle$ and $|1\rangle$ defined in the next section.

A Qubit

For $j$ in $\{0, 1\}$, the computational basis ket $|j\rangle$ of the qubit is the column vector of $\mathbb{C}^2$ with entry $1$ in position $j$ and $0$ in the other: $$|0\rangle = \begin{pmatrix} 1 \\ 0 \end{pmatrix}, \qquad |1\rangle = \begin{pmatrix} 0 \\ 1 \end{pmatrix}.$$(1)^✦

The Pauli matrices $X$, $Y$ and $Z$ are the complex $2 \times 2$ matrices $$X = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}, \qquad Y = \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix}, \qquad Z = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix},$$(2) with $i$ the imaginary unit.

For a vector $|v\rangle$ of $\mathbb{C}^2$ with components $v_0$ and $v_1$, the outer product $|v\rangle\langle v|$ is the complex $2 \times 2$ matrix with entries $$\bigl( |v\rangle\langle v| \bigr)_{jk} = v_j \, v_k^{*},$$(3)^✦ where $j, k \in \{0, 1\}$ and $v_k^{*}$ is the complex conjugate of $v_k$.

Four States

Four more states of the qubit are built from the complex number $$\frac{1}{\sqrt{2}}.$$(4)^✦

The states $|+\rangle$ and $|-\rangle$ of the qubit are the vectors of $\mathbb{C}^2$ with amplitude $1/\sqrt{2}$ on $|0\rangle$ and, on $|1\rangle$, amplitude $1/\sqrt{2}$ and $-1/\sqrt{2}$ respectively: $$|+\rangle = \begin{pmatrix} 1/\sqrt{2} \\ 1/\sqrt{2} \end{pmatrix},$$(5) $$|-\rangle = \begin{pmatrix} 1/\sqrt{2} \\ -1/\sqrt{2} \end{pmatrix}.$$(6)

The states $|+i\rangle$ and $|-i\rangle$ of the qubit are the vectors of $\mathbb{C}^2$ with amplitude $1/\sqrt{2}$ on $|0\rangle$ and, on $|1\rangle$, amplitude $i/\sqrt{2}$ and $-i/\sqrt{2}$ respectively: $$|+i\rangle = \begin{pmatrix} 1/\sqrt{2} \\ i/\sqrt{2} \end{pmatrix},$$(7) $$|-i\rangle = \begin{pmatrix} 1/\sqrt{2} \\ -i/\sqrt{2} \end{pmatrix}.$$(8)

The Claim

What Is Checked

The claim is stated over the Pauli matrices (2), the computational basis kets (1), the four states (5) to (8) with their constant (4), and the outer product (3), all of them explicit complex matrices and vectors of $\mathbb{C}^2$.

About these constructions the library proves the four eigenvalue equations and the three diagonal representations of the theorem, (9) to (15).