The Exercise

Two spin $\tfrac{1}{2}$ particles have a total spin, with one component along each of the three axes, each an operator on the state space of the pair. The exercise is to show that these three components obey the commutation relations of an angular momentum, in units with $\hbar = 1$.

Two Qubits

Each particle is a qubit, with state space $\mathbb{C}^2$, and the pair has state space $\mathbb{C}^2 \otimes \mathbb{C}^2$. The Pauli matrices $\sigma_1, \sigma_2, \sigma_3$ are the complex $2 \times 2$ matrices of the operators $X$, $Y$ and $Z$, one for each axis: $$\sigma_1 = X = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}, \qquad \sigma_2 = Y = \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix}, \qquad \sigma_3 = Z = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}.$$(1)

The total spin $j_k$ of the two qubits along the axis $k$ is the operator on $\mathbb{C}^2 \otimes \mathbb{C}^2$ that adds the spin $\sigma_k / 2$ of the first qubit to that of the second, with $I$ the $2 \times 2$ identity: $$j_k = \frac{1}{2} \left( \sigma_k \otimes I + I \otimes \sigma_k \right).$$(2)^✦

The Claim

The Levi-Civita symbol $\varepsilon_{klm}$ is the totally antisymmetric integer symbol in three indices, each running over $1, 2, 3$: it is $+1$ when $(k, l, m)$ is a cyclic permutation of $(1, 2, 3)$, $-1$ when it is an anticyclic one, and $0$ when two of its indices coincide.^✦

What Is Checked

The claim is stated over the total spin components $j_1, j_2, j_3$ of (2), built from the Pauli matrices (1), and over the Levi-Civita symbol. About these constructions the library proves (3) for all nine pairs of axes.