Improved finite-size key rates for discrete-modulated continuous-variable quantum key distribution under coherent attacks

Continuous-variable quantum key distribution (CVQKD) with discrete modulation combines advantages of CVQKD, such as implementability using readily available technologies, with advantages of discrete variable quantum key distribution, such as easier error correction procedures. We consider a prepare-and-measure CVQKD protocol, where Alice chooses from a set of four coherent states and Bob performs a heterodyne measurement, the result of which is discretized in both key and test rounds. We provide a security proof against coherent attacks in the finite-size regime based on the generalized entropy accumulation theorem. We compute the corresponding asymptotic key rate imposing a photon-number cutoff and recent advances in conic optimization. Our method yields significant improvements on key rates: at metropolitan distances, it provides positive key rates for block sizes of the order of 10^8 rounds, orders of magnitude smaller compared to previous works.