Practical Framework for Analyzing High-Dimensional Quantum Key Distribution Setups

High-dimensional (HD) entanglement promises both enhanced key rates and overcoming obstacles faced by modern-day quantum communication. However, modern convex optimization-based security arguments are limited by computational constraints; thus, accessible dimensions are far exceeded by progress in HD photonics, bringing forth a need for efficient methods to compute key rates for large encoding dimensions. In response to this problem, we present a flexible analytic framework facilitated by the dual of a semidefinite program and diagonalizing operators inspired by entanglement-witness theory, enabling the efficient computation of key rates in high-dimensional systems. To facilitate the latter, we show how matrix completion techniques can be incorporated to effectively yield improved, computable bounds on the key rate in paradigmatic high-dimensional systems of time- or frequency-bin entangled photons and beyond, revealing the potential for very high-dimensions to surpass low dimensional protocols already with existing technology. In our accompanying work, (F. Kanitschar and M. Huber, Composable finite-size security of high-dimensional quantum key distribution protocols), available on arXiv, we show how our findings can be used to establish finite-size security against coherent attacks for general HD-QKD protocols both in the fixed- and variable-length scenario.