Directional amplification, in which signals are amplified selectively depending on their propagation direction, is a key resource for quantum information processing and stands in one-to-one correspondence with non-trivial non-Hermitian topology. So far, this correspondence has concerned the mean fields, and thus classical response. Here we turn to the quantum fluctuations, giving access to correlations and entanglement. We show that non-trivial topology produces correlations that grow exponentially with the distance between modes and approach the largest values compatible with the uncertainty relations. The associated correlation length diverges at the topological phase transition. Entanglement nonetheless remains local, set by the competition between correlations and the asymmetry of the mode occupations. This prepares the ground for exploring the quantum properties of non-Hermitian topological systems with state-of-the-art platforms such as cavity optomechanics and superconducting circuits.
