Finite-key analysis for time-energy high-dimensional quantum key distribution

Time-energy high-dimensional quantum key distribution (HD-QKD) leverages the high-dimensional nature of time-energy entangled biphotons and the loss tolerance from the discrete nature of single-photon detection thus enabling long-distance key distribution with high photon information efficiency (PIE). However, the security of HD-QKD has only been proven in the asymptotic regime, or in the finite-key regime against a limited set of attacks. Here we fill this gap by providing a rigorous HD-QKD security proof for general (coherent) attacks in the finite-key regime. The proof follows from a novel uncertainty relation for time and conjugatetime measurements, which bounds the secure information, and an efficient decoy-state protocol for parameter estimation. For the first time, we prove the feasibility of realizing secure and composable HD-QKD against the most powerful eavesdropping attacks across metropolitan-area distances.