Efficiency in Quantum Key Distribution Protocols with Entangled Gaussian States

Quantum key distribution (QKD) refers to specific quantum strategies which permit the secure distribution of a secret key between two parties that wish to communicate secretly. Quantum cryptography has proven unconditionally secure in ideal scenarios and has been successfully implemented using quantum states with finite (discrete) as well as infinite (continuous) degrees of freedom. Here, we analyze the efficiency of QKD protocols that use as a resource entangled gaussian states and gaussian operations only. In this framework, it has already been shown that QKD is possible (M. Navascu\'es et al. Phys. Rev. Lett. 94, 010502 (2005)) but the issue of its efficiency has not been considered. We propose a figure of merit (the efficiency $E$) to quantify the number of classical correlated bits that can be used to distill a key from a sample of $N$ entangled states. We relate the efficiency of the protocol to the entanglement and purity of the states shared between the parties.