High-dimensional quantum key distribution rates for multiple measurement bases

We investigate the advantages of high-dimensional encoding for a quantum key distribution protocol. In particular, we address a BBM92-like protocol where the dimension of the systems can be larger than two and more than two mutually unbiased bases (MUBs) can be employed. Indeed, it is known that, for a system whose dimension $d$ is a prime or the power of a prime, up to $d+1$ MUBs can be found. We derive an analytic expression for the asymptotic key rate when $d+1$ MUBs are exploited and show the effects of using different numbers of MUBs on the performance of the protocol. Then, we move to the non-asymptotic case and optimize the finite key rate against collective and coherent attacks for generic dimension of the systems and all possible numbers of MUBs. In the finite-key scenario, we find that, if the number of rounds is small enough, the highest key rate is obtained by exploiting three MUBs, instead of $d+1$ as one may expect.