The Exercise

The exercise is about the three Pauli matrices $X$, $Y$ and $Z$ of a qubit and their commutators, the commutator of two matrices $A$ and $B$ being $[A, B] = AB - BA$. It asks for the commutator of each pair of Pauli matrices, and then for one formula that gives all of them at once, in terms of the Pauli matrices numbered $1$, $2$, $3$ and the Levi-Civita symbol, the antisymmetric symbol on three indices.

The Pauli Matrices

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

The Pauli matrices $\sigma_j$, for $j = 1, 2, 3$, are the components of the Pauli vector $\vec{\sigma}$, the same three matrices taken in this order: $$\sigma_1 = X, \qquad \sigma_2 = Y, \qquad \sigma_3 = Z.$$(2)^✦

The Levi-civita Symbol

The Levi-Civita symbol $\varepsilon_{jkl}$ is the integer symbol on three indices $j$, $k$, $l$ from $\{1, 2, 3\}$ that vanishes whenever two of the indices coincide and on the six permutations of $(1, 2, 3)$ takes the values $$\varepsilon_{123} = \varepsilon_{231} = \varepsilon_{312} = 1, \qquad \varepsilon_{132} = \varepsilon_{213} = \varepsilon_{321} = -1.$$(3)^✦

The Claim

What Is Checked

The claim is stated over the matrices $X$, $Y$ and $Z$ written out entry by entry in (1), their numbering as $\sigma_1$, $\sigma_2$, $\sigma_3$ in (2), and the integer values of the Levi-Civita symbol in (3); the commutator is the matrix commutator $AB - BA$.

About these matrices the library proves the four identities of the theorem as equalities of complex $2 \times 2$ matrices: (4), (5) and (6) for $X$, $Y$ and $Z$ themselves, and (7) for each of the nine ordered pairs $(j, k)$.