The Exercise

Two qubits have the state space $\mathbb{C}^2 \otimes \mathbb{C}^2$, with the computational basis $|00\rangle$, $|01\rangle$, $|10\rangle$, $|11\rangle$. A Pauli operator $A$ on the first qubit and a Pauli operator $B$ on the second give the operator $A \otimes B$ on the pair. The exercise asks for three such operators as $4 \times 4$ matrices in that basis: $I \otimes X$ and $X \otimes I$, which apply $X$ to the second qubit alone and to the first alone, and $X \otimes Z$. It then asks whether the tensor product is commutative, whether $A \otimes B$ and $B \otimes A$ are always the same operator.

The Matrices

The Pauli matrices $I$, $X$ and $Z$ of a qubit are the complex $2 \times 2$ matrices, with rows and columns labelled by $0$ and $1$, $$I = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}, \qquad X = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}, \qquad Z = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}.$$(1)

The tensor product $A \otimes B$ of two $2 \times 2$ matrices $A$ and $B$ is the matrix whose rows are labelled by pairs $(j,k)$ and whose columns are labelled by pairs $(j',k')$ of indices, the first for the first factor and the second for the second, with entries $$(A \otimes B)_{(j,k),(j',k')} = A_{jj'}\, B_{kk'} .$$(2)

The $4 \times 4$ matrix $[M]$ of a complex matrix $M$ whose rows and columns are labelled by pairs $(j,k)$ of indices $0$ and $1$ is the matrix with rows and columns labelled $0$ to $3$, the pair $(j,k)$ becoming the single index $2j + k$: $$[M]_{2j+k,\; 2j'+k'} = M_{(j,k),(j',k')} .$$(3)^✦

The Claim

What Is Checked

The claim is stated over the Pauli matrices (1), the tensor product of matrices (2), and the $4 \times 4$ matrix (3) of a matrix labelled by pairs of indices.