  Quantum key distribution (QKD) allows two remote users to establish a secret
key in the presence of an eavesdropper. The users share quantum states prepared
in two mutually-unbiased bases: one to generate the key while the other
monitors the presence of the eavesdropper. Here, we show that a general
$d$-dimension QKD system can be secured by transmitting only a subset of the
monitoring states. In particular, we find that there is no loss in the secure
key rate when dropping one of the monitoring states. Furthermore, it is
possible to use only a single monitoring state if the quantum bit error rates
are low enough. We apply our formalism to an experimental $d=4$ time-phase QKD
system, where only one monitoring state is transmitted, and obtain a secret key
rate of $17.4 \pm 2.8$ Mbits/s at a 4 dB channel loss and with a quantum bit
error rate of $0.045\pm0.001$ and $0.044\pm0.001$ in time and phase bases,
respectively, which is 58.4 % of the secret key rate that can be achieved with
the full setup. This ratio can be increased, potentially up to 100 %, if the
error rates in time and phase basis are reduced. Our results demonstrate that
it is possible to substantially simplify the design of high-dimension QKD
systems, including those that use the spatial or temporal degrees-of-freedom of
the photon, and still outperform qubit-based ($d = 2$) protocols.
