Quantum summation using d-level entanglement swapping

Abstract

In this paper, we propose a secure multi-party quantum modulo-d summation protocol using d-level entanglement swapping, where a malicious but non-collusive third party (TP) who plays a role of entanglement distribution between the participants is involved. TP is assumed to be malicious but non-collusive, which means TP can launch various attacks within boundaries of quantum mechanics except the collusion with any dishonest participant. Our protocol is congenitally free from Trojan horse attacks because there is no need to relay encoded states. In addition, only d-level Bell measurement is required for the users except TP, which makes our protocol’s implementation more feasible. We show that the presented protocol is secure against outside as well as participant attacks. Second, we give a detailed comparison of different quantum modulo-d summation protocols, in terms of qudit cost, quantum resources, and quantum operations. In the end, we discuss how to replace cat states with d-level Bell states in Je et al.’s protocols (Quantum Inf Process 18(6):168, 2019) for easier implementation with current quantum technology.