  We report experimental verification of the commutation relation for Pauli
spin operators using quantum interference of the single-photon polarization
state. By superposing the quantum operations $\sigma_z \sigma_x$ and $\sigma_x
\sigma_z$ on a single-photon polarization state, we have experimentally
implemented the commutator, $[\sigma_{z}, \sigma_{x}]$, and the anticommutator,
$\{\sigma_{z}, \sigma_{x}\}$, and have demonstrated the relative phase factor
of $\pi$ between $\sigma_z \sigma_x$ and $\sigma_x \sigma_z$ operations. The
experimental quantum operation corresponding to the commutator, $[\sigma_{z},
\sigma_{x}]=k\sigma_y$, showed process fidelity of 0.94 compared to the ideal
$\sigma_y$ operation and $|k|$ is determined to be $2.12\pm0.18$.
