Explicit asymptotic secret key rate of continuous-variable quantum key distribution with an arbitrary modulation

We establish an analytical lower bound on the asymptotic secret key rate of continuous-variable quantum distribution with arbitrary modulation coherent states. Previously, such bounds were only available for protocols a Gaussian modulation, and numerical existed in case simple phase-shift-keying modulations. The latter obtained as solution convex optimization problems our new matches results Ghorai et al. (2019), up to precision. more relevant quadrature amplitude (QAM) could not be analyzed previous techniques, due their large number Our shows that relatively small constellation sizes, say 64 states, are essentially sufficient obtain performance close true therefore attractive large-scale deployment distribution. also derive similar when consists necessarily pure.