  Quantum coherences characterise the ability of particles to quantum
mechanically interfere within some given distances. In the context of noisy
many-body quantum systems these coherences can fluctuate. A simple toy model to
study such fluctuations in an out-of-equilibrium setting is the open quantum
symmetric simple exclusion process (Q-SSEP) which describes spinless fermions
in one dimension hopping to neighbouring sites with random amplitudes coupled
between two reservoirs. Here we show that the dynamics of fluctuations of
coherences in Q-SSEP have a natural interpretation as free cumulants, a concept
from free probability theory. Based on this insight we provide heuristic
arguments why we expect free probability theory to be an appropriate framework
to describe coherent fluctuations in generic mesoscopic systems where the noise
emerges from a coarse-grained description. In the case of Q-SSEP we show how
the link to free probability theory can be used to derive the time evolution of
connected fluctuations of coherences as well as a simple steady state solution.
