Quantum Key Distribution with Nonideal Heterodyne Detection: Composable Security of Discrete-Modulation Continuous-Variable Protocols

Continuous-variable quantum key distribution exploits coherent measurements of the electromagnetic field, i.e., homodyne or heterodyne detection. The most advanced security proofs developed so far have relied on idealized mathematical models for such measurements, which assume that the measurement outcomes are continuous and unbounded variables. As physical-measurement devices have a finite range and precision, these mathematical models only serve as an approximation. It is expected that, under suitable conditions, the predictions obtained using these simplified models will be in good agreement with the actual experimental implementations. However, a quantitative analysis of the error introduced by this approximation, and of its impact on composable security, have been lacking so far. Here, we present a theory to rigorously account for the experimental limitations of realistic heterodyne detection. We focus on collective attacks and present security proofs for the asymptotic and finite-size regimes, the latter being within the framework of composable security. In doing this, we establish for the first time the composable security of discrete-modulation continuous-variable quantum key distribution in the finite-size regime. Tight bounds on the key rates are obtained through semidefinite programming and do not rely on a truncation of the Hilbert space.