  The six-state protocol is a discrete-variable protocol for quantum key
distribution, that permits to tolerate a noisier channel than the BB84
protocol. In this work we provide a lower bound on the maximum achievable key
rate of a practical implementation of the entanglement-based version of the
six-state protocol. Regarding the experimental set-up we consider that the
source is untrusted and the photon-number statistics is measured using
photon-number-resolving detectors. We provide the formula for the key rate for
a finite initial number of resources. As an illustration of the considered
formalism, we calculate the key rate for the setting where the source produces
entangled photon pairs via parametric down-conversion and the losses in the
channel depend on the distance. As a result we find that the finite-key
corrections for the considered scenario are not negligible and they should be
considered in any practical analysis.

Quantum Physics