  We experimentally realize a nonlinear quantum protocol on single-photon
qubits with linear optical elements and appropriate measurements. The quantum
nonlinearity is induced by post-selecting the polarization qubit based on a
measurement result obtained on the spatial degree of freedom of the single
photon which plays the role of a second qubit. Initially, both qubits are
prepared in the same quantum state and an appropriate two-qubit unitary
transformation entangles them before the measurement on the spatial part. We
analyze the result by quantum state tomography on the polarization degree of
freedom. We then demonstrate the usefulness of the protocol for quantum state
discrimination by iteratively applying it on either one of two slightly
different quantum states which rapidly converge to different orthogonal states
by the iterative dynamics. Our work opens the door to employ effective quantum
nonlinear evolution for quantum information processing.
