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 $σ_z σ_x$ and $σ_x σ_z$ on a single-photon polarization state, we have experimentally implemented the commutator, $[σ_{z}, σ_{x}]$, and the anticommutator, $\{σ_{z}, σ_{x}\}$, and have demonstrated the relative phase factor of $π$ between $σ_z σ_x$ and $σ_x σ_z$ operations. The experimental quantum operation corresponding to the commutator, $[σ_{z}, σ_{x}]=kσ_y$, showed process fidelity of 0.94 compared to the ideal $σ_y$ operation and $|k|$ is determined to be $2.12\pm0.18$.

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