  Entanglement properties of driven quantum systems can potentially differ from
the equilibrium situation due to long range coherences. We confirm this
observation by studying a suitable toy model for mesoscopic transport~: the
open quantum symmetric simple exclusion process (QSSEP). We derive exact
formulae for its mutual information between different subsystems in the steady
state and show that it satisfies a volume law. Surprisingly, the QSSEP
entanglement properties only depend on data related to its transport properties
and we suspect that such a relation might hold for more general mesoscopic
systems. Exploiting the free probability structure of QSSEP, we obtain these
results by developing a new method to determine the eigenvalue spectrum of
sub-blocks of random matrices from their so-called local free cumulants -- a
mathematical result on its own with potential applications in the theory of
random matrices. As an illustration of this method, we show how to compute
expectation values of observables in systems satisfying the Eigenstate
Thermalization Hypothesis (ETH) from the local free cumulants.
