  Computing the key rate in quantum key distribution (QKD) protocols is a long
standing challenge. Analytical methods are limited to a handful of protocols
with highly symmetric measurement bases. Numerical methods can handle arbitrary
measurement bases, but either use the min-entropy, which gives a loose lower
bound to the von Neumann entropy, or rely on cumbersome dedicated algorithms.
Based on a recently discovered semidefinite programming (SDP) hierarchy
converging to the conditional von Neumann entropy, used for computing the
asymptotic key rates in the device independent case, we introduce an SDP
hierarchy that converges to the asymptotic secret key rate in the case of
characterised devices. The resulting algorithm is efficient, easy to implement
and easy to use. We illustrate its performance by recovering known bounds on
the key rate and extending high-dimensional QKD protocols to previously
intractable cases. We also use it to reanalyse experimental data to demonstrate
how higher key rates can be achieved when the full statistics are taken into
account.
