Measurement-device-independent quantum key distribution with hyper-encoding

Measurement device-independent quantum key distribution (MDI-QKD) protocols are immune to all possible attacks on the photon detectors during quantum communication, but their key generation rates are low compared with those of other QKD schemes. Increasing each individual photon's channel capacity is an e fficient way to increase the key generation rate, and high-dimensional (HD) encoding is a powerful tool for increasing the channel capacity of photons. In this paper, we propose an HD MDI-QKD protocol with qudits hyper-encoded in spatial mode and polarization degrees of freedom (DOFs). In the proposed protocol, keys can be generated using the spatial mode and polarization DOFs simultaneously. The proposed protocol is unconditionally secure, even for weak coherent pulses with decoy states. The proposed MDI-QKD protocol may be useful for future quantum secure communication applications.