Generation of four-photon polarization entangled decoherence-free states with cross-Kerr nonlinearity

We propose a theoretical protocol for preparing four-photon polarization entangled decoherence-free states, which are immune to the collective noise. With the assistance of the cross-Kerr nonlinearities, a two-photon spatial entanglement gate, two controlled-NOT gates, a four-photon polarization entanglement gate are inserted into the circuit, where X homodyne measurements are aptly applied. Combined with some swap gates and simple linear optical elements, four-photon polarization entangled decoherence-free states which can be utilized to represent two logical qubits, |0〉 L and |1〉 L are achieved at the output ports of the circuit. This generation scheme may be implemented with current experimental techniques.