Coalescent entanglement and the conditional dependence of the times to common ancestry of mutually exclusive pairs of individuals

The Kingman coalescent is a continuous-time diffusion approximation of the times to common ancestry of a sample of individuals drawn from a Wright-Fisher population. Here, we use the coalescent to answer a simple question: if we know the ancestry of 2 randomly sampled individuals in the population, what does it tell us about the ancestry of 2 other randomly sampled individuals? We show that there is a conditional dependency between the times to common ancestry between pairs of randomly sampled individuals. We call this "coalescent entanglement," and we demonstrate its effects through simulation. The effects of entanglement extend beyond the coalescent to phylogenetic birth-death processes in general. Entanglement also exerts its effects when the pairs of individuals chosen share no common lineages in the paths that connect the individuals in each pair.