Exact computation of tail probability of hypergeometric distribution and its application to Quantum Key Distribution

Abstract

Tomamichel et al. improved a key rate of a Quantum Key Distribution (QKD) protocol with finite number of qubits. To compute the key rate, we need to estimate a phase error rate, which obey a hypergeometric distribution. They did not compute directly the tail probability of the hypergeometric distribution to obtain the estimate of the error rate, and they used its upper bound instead. We improve the key rate of the QKD by exact numerical computation of the tail probability. We also propose a method to decrease the computation time for exact numerical computation.