Experimental verification of the commutation relation for Pauli spin operators using single-photon quantum interference

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 $2.12\pm0.18$.