Security of deterministic key distribution with higher-dimensional systems

We determine the security aspects of the two-way quantum key distribution protocol designed by M. Lucamarini and S. Mancini in 2005 (in short, the LM05 protocol) by employing arbitrary finite-dimensional systems against both individual and collective attacks by the eavesdropper. We explicitly delineate the constituent steps in higher dimensions using the Heisenberg-Weyl operators for the encoding and the measurement operations. For individual attacks, we consider cloning operations by the eavesdropper and demonstrate a dimensional advantage where secret keys can be generated for greater strengths of interception. To analyze security under collective attacks, we employ a purification scheme and derive the key rate using entropic uncertainty relations. Further, we exhibit how the protocol is more robust against eavesdropping with increasing dimension of the systems used, and compare the performance with that of the entangled two-way secure dense coding protocol when affected by correlated and uncorrelated noise, which models the presence of the eavesdropper.