A probabilistic CNOT gate for coherent state qubits

We propose a scheme for implementing a probabilistic controlled-NOT (CNOT) gate for coherent state qubits using only linear optics and a particular four-mode state. The proposed optical setup works, as a CNOT gate, near-faithful when |α|2⩾25 and independent of the input state. The key element for realizing the proposed CNOT scheme is the entangled four-mode state.