Finite-Size Security for Discrete-Modulated Continuous-Variable Quantum Key Distribution Protocols

Discrete-modulated (DM) continuous-variable quantum key distribution (CV-QKD) protocols are promising candidates for commercial implementations of quantum communication networks due to their experimental simplicity. While tight security analyses in the asymptotic limit exist, proofs in the finitesize regime are still subject to active research. We present a composable finite-size security proof against independently and identically distributed collective attacks for a general DM CV-QKD protocol. We introduce a new energy testing theorem to bound the effective dimension of Bob's system and rigorously prove security within Renner's epsilon-security framework and address the issue of acceptance sets in protocols and their security proof. We want to highlight that our method also allows for nonunique acceptance statistics, which is necessary in practise. Finally, we extend and apply a numerical security proof technique to calculate tight lower bounds on the secure key rate. To demonstrate our method, we apply it to a quadrature phase-shift keying protocol for both untrusted, ideal and trusted, nonideal detectors. The results show that our security proof method yields secure finite-size key rates under experimentally viable conditions up to at least 72 km transmission distance.