Finite key analysis for symmetric attacks in quantum key distribution

Abstract

We introduce a constructive method to calculate the achievable secret key rate for a generic class of quantum key distribution protocols, when only a finite number $n$ of signals is given. Our approach is applicable to all scenarios in which the quantum state shared by Alice and Bob is known. In particular, we consider the six state protocol with symmetric eavesdropping attacks, and show that for a small number of signals, i.e., below $n\ensuremath{\sim}{10}^{4}$, the finite key rate differs significantly from the asymptotic value for $n\ensuremath{\rightarrow}\ensuremath{\infty}$. However, for larger $n$, a good approximation of the asymptotic value is found. We also study secret key rates for protocols using higher-dimensional quantum systems.