Finite size analysis of measurement device independent quantum cryptography with continuous variables

We study the impact of finite-size effects on the key rate of continuous-variable (CV) measurement-device-independent (MDI) quantum key distribution (QKD). Inspired by the parameter estimation technique developed in [Rupert $\mathit{\text{et}}\mathit{\text{al.}}$ Phys. Rev. A $\mathbf{\text{90}}$, 062310 (2014)]~we adapt it to study CV-MDI-QKD and, assuming realistic experimental conditions, we analyze the impact of finite-size effects on the key rate. We find that, increasing the block-size, the performance of the protocol converges towards the ideal one, and that block-sizes between ${10}^6$ and ${10}^9$ data points can already provide a key rate $\sim {10}^{-2}$ bit/use over metropolitan distances.