The quantum discrimination of two non-coherent states draws much attention recently. In this letter, we first consider the quantum discrimination of two noiseless displaced number states. Then we derive the Fock representation of noisy displaced numb
er states and address the problem of discriminating between two noisy displaced number states. We further prove that the optimal quantum discrimination of two noisy displaced number states can be achieved by the Kennedy receiver with threshold detection. Simulation results verify the theoretical derivations and show that the error probability of on-off keying modulation using a displaced number state is significantly less than that of on-off keying modulation using a coherent state with the same average energy.
