Hong–Ou–Mandel interference in quantum optics, monogamy of entanglement, nonorthogonality, and untrusted nodes

Quantum key distribution systems with an untrusted intermediate node described by the so-called measurement device independent (MDI) protocol have been actively studied in the last decade. In early works, it was only argued why such a quantum key distribution system ensures the security of distributed keys mentioning that the security proof of the MDI protocol, which was not presented, is similar to that for the basic Bennett–Brassard 84 (BB84) protocol. For this reason, despite the existing experimental implementations of the MDI quantum key distribution system, physical reasons for the protocol security are still questionable. Such quantum key distribution systems provide a common key between two network nodes connected through the intermediate untrusted node, which does not require protection of the equipment on it, and an eavesdropper sees the entire operation of the equipment, including the results of the operation of photodetectors. In this work, the MDI protocol has been analyzed. It has been shown that the physical reasons for the protocol security are based on fundamental properties such as the interference of photons from different sources, monogamy of entanglement, and nonorthogonality of states. A simple and explicit derivation is given showing the equivalence of the MDI and BB84 protocols and physical reasons for the identity of the corresponding expressions for the length of the final key.