Efficient quantum key distribution over a collective noise channel

We present two efficient quantum key distribution schemes over two different collective-noise channels. The accepted hypothesis of collective noise is that photons travel inside a time window small compared to the variation of noise. Noiseless subspaces are made up of two Bell states and the spatial degree of freedom is introduced to form two nonorthogonal bases. Although these protocols resort to entangled states for encoding the key bit, the receiver is only required to perform single-particle product measurements and there is no basis mismatch. Moreover, the detection is passive as the receiver does not switch his measurements between two conjugate measurement bases to get the key.