  The motivation for this thesis is to find a fluctuating hydrodynamic
description of quantum coherent effects in mesoscopic quantum systems with
diffusive transport properties. Coherent effects are inscribed into the
coherences (two-point Green's function) on which we focus as the building
blocks of such a theory. Our approach is rather mathematical and not related to
concrete experiments. We study the quantum symmetric simple exclusion process
(QSSEP), a potentially iconic model describing transport of noisy free fermions
on a 1D lattice, which has the minimal structure to capture what we are
interested in: Long-ranged coherences and diffusive transport. In mean, QSSEP
reduces to the symmetric simple exclusion process (SSEP), a classical toy model
that was important in the development of the macroscopic fluctuation theory, a
fluctuating hydrodynamic description of diffusive transport in classical
systems. Studying QSSEP we hope to make progress towards a quantum coherent
extension of the macroscopic fluctuation theory.
  The thesis summarizes many results we have obtained about QSSEP in the last
three years, such as the dynamical equation and an exact stationary solution
for correlation functions of coherences at hydrodynamic scales, the
distribution of entanglement in QSSEP and an argument why QSSEP might be an
effective noisy description for more generic mesoscopic quantum systems. Many
of these results are due to a relation between the statistical properties of
coherences in QSSEP and free probability theory. We devote a whole chapter to
this relation and present, as a by-product, a method to characterize the
spectrum of subblocks of a large class of structured random matrices.
