  In quantum information theory, quantum discord has been proposed as a tool to
characterise the presence of "quantum correlations" between the subparts of a
given system. Whether a system behaves quantum-mechanically or classically is
believed to be impacted by the phenomenon of decoherence, which originates from
the unavoidable interaction between this system and an environment.
Generically, decoherence is associated with a decrease of the state purity,
i.e. a transition from a pure to a mixed state. In this paper, we investigate
how quantum discord is modified by this quantum-to-classical transition. This
study is carried out on systems described by quadratic Hamiltonians and
Gaussian states, with generalised squeezing parameters. A generic
parametrisation is also introduced to describe the way the system is
partitioned into two subsystems. We find that the evolution of quantum discord
in presence of an environment is a competition between the growth of the
squeezing amplitude and the decrease of the state purity. In phase space, this
corresponds to whether the semi-minor axis of the Wigner ellipse increases or
decreases, which has a clear geometrical interpretation. Finally, these
considerations are applied to primordial cosmological perturbations, thus
allowing us to investigate how large-scale structures in our universe, which
are believed to arise from quantum fluctuations, can exhibit classical
properties.
