Tight bound on the coherent-state quantum key distribution with heterodyne detection

We propose an upper bound for the eavesdropper's information in the direct and reverse reconciliated coherent states quantum key distribution protocols with heterodyne detection. This bound is derived by maximizing the leaked information over the symplectic group of transformations that spans every physical Gaussian attack on individual pulses. We exhibit four different attacks that reach this bound, which shows that this bound is tight. Finally, we compare the secret key rate obtained with this bound to the homodyne rate.