  Time-energy high-dimensional quantum key distribution (HD-QKD) leverages the
high-dimensional nature of time-energy entangled biphotons and the loss
tolerance of single-photon detection to achieve long-distance key distribution
with high photon information efficiency. To date, the general-attack security
of HD-QKD has only been proven in the asymptotic regime, while HD-QKD's
finite-key security has only been established for a limited set of attacks.
Here we fill this gap by providing a rigorous HD-QKD security proof for general
attacks in the finite-key regime. Our proof relies on a novel entropic
uncertainty relation that we derive for time and conjugate-time measurements
using dispersive optics, and our analysis includes an efficient decoy-state
protocol in its parameter estimation. We present numerically-evaluated
secret-key rates illustrating the feasibility of secure and composable HD-QKD
over metropolitan-area distances when the system is subjected to the most
powerful eavesdropping attack.
