The Exercise

The state space of a qubit is $\mathbb{C}^2$, with the orthonormal computational basis $|0\rangle$, $|1\rangle$. The Pauli matrices $I$, $X$, $Y$ and $Z$ are operators on this space, given by their matrices in that basis. The exercise asks for each of them in outer product notation, as a linear combination of the four operators $|i\rangle\langle j|$ with $i$ and $j$ each $0$ or $1$.

The Computational Basis

The computational basis vector $|i\rangle$, for $i$ equal to $0$ or $1$, is the column vector in $\mathbb{C}^2$ with entry $1$ in position $i$ and $0$ in the other position: $$|0\rangle = \begin{pmatrix} 1 \\ 0 \end{pmatrix}, \qquad |1\rangle = \begin{pmatrix} 0 \\ 1 \end{pmatrix}.$$(1)^✦

The outer product $|i\rangle\langle j|$ of two basis vectors is the complex $2 \times 2$ matrix obtained by multiplying the column $|i\rangle$ by the row $\langle j|$, the conjugate transpose of $|j\rangle$.^✦ Its entry in row $k$ and column $l$ is, with $\delta_{ab}$ the Kronecker delta, equal to $1$ if $a=b$ and $0$ otherwise, $$\bigl(|i\rangle\langle j|\bigr)_{kl} = (|i\rangle)_k \, \overline{(|j\rangle)_l} = \delta_{ik}\,\delta_{jl}.$$(2)

The Pauli Matrices

The Pauli identity $I$ is the $2 \times 2$ identity matrix, $$I = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix},$$(3) and the Pauli matrices $X$, $Y$ and $Z$ are the complex $2 \times 2$ matrices $$X = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}, \qquad Y = \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix}, \qquad Z = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix},$$(4) where the $i$ in $Y$ is the imaginary unit.

The Claim

By (2), the coefficient of $|i\rangle\langle j|$ in the expansion of a $2 \times 2$ matrix is the entry of that matrix in row $i$ and column $j$.

What Is Checked

The claim is stated over the two computational basis vectors (1), their outer products (2), and the four matrices $I$, $X$, $Y$ and $Z$ of (3) and (4). The library proves the four identities (5) to (8) about these constructions, each as an equality of complex $2 \times 2$ matrices.