Resource-saving quantum key distribution based on three-photon matrix product states

Quantum key distribution (QKD) protocols based on entangled states published so far do not fully exploit the potential of coefficients (i.e., the probability amplitude) of entangled states in the generation of cryptographic keys. In contrast, a matrix product state (MPS), which is a type of the well-defined tensor network states based on entanglement for quantum many-body systems, provides a complete description of the entire state entanglement. Unlike the von Neumann entropy, which describes the “quantity” of entanglements, MPS describes a specific entanglement relationship among states. In this paper, we explore an application of MPS for QKD protocols. We expect that the MPS representation allows us to design resource-saving QKD protocols for a smaller number of transmitted entangled photons than in the traditional QKD protocols which do not use MPS, because a qubit is a costly quantum resource. We show how to design a two-party (i.e., Alice and Bob) QKD protocol based on a three-photon entangled state that can be completely represented by three-photon MPS. In our protocol, Alice transforms the second entangled photons of three-photon MPS to Bob. A special structure of three-photon MPS allows not only to increase efficiency of our protocol but also to share keys amongst subsets of MPS. Our protocol illustrates the new advantages of applying three-photon MPS for QKD protocols. They go well beyond efficiency gain by halving the number of entangled photons used in QKD protocols.