Fault-tolerant measurement-device-independent quantum key distribution in a decoherence-free subspace

Two modified measurement-device-independent quantum key distribution protocols based on the decoherence-free subspace are presented in this study. The proposed protocols are tolerant of the fault with collective-rotation noise and collective-dephasing noise. Exploiting the logical qubits comprised by two pairs of entanglement photons in decoherence-free subspace states, the mutually unbiased bases are formed by introducing the spatial degrees of freedom which reduces the experiment difficulty. There are only Bell-state preparation and collective Bell-state measurement needed in our protocols. Moreover, a brief discussion on the security of the proposal in the communication process is given.