The Exercise

A single qubit has the computational basis $|0\rangle$, $|1\rangle$, in which an operator on the qubit is written as a complex $2 \times 2$ matrix. The exercise asks to show that two of the Pauli matrices, written $X$ and $Y$ below, have trace zero: for each, the sum of its two diagonal entries vanishes.

The Pauli Matrices

The Pauli $X$ matrix is the complex $2 \times 2$ matrix $$X = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}.$$(1)^✦

The Pauli $Y$ matrix is the complex $2 \times 2$ matrix $$Y = \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix},$$(2)^✦ with $i$ the imaginary unit.

The Claim

What Is Checked

The claim is stated over the two matrices (1) and (2), with complex entries, and takes no hypothesis.