  The flavor evolution of neutrinos in environments with large neutrino number
densities is an open problem at the nexus of astrophysics and neutrino flavor
physics. Among the many unanswered questions pertaining to this problem, it
remains to be determined whether neutrino-neutrino coherent scattering can give
rise to nontrivial quantum entanglement among neutrinos, and whether this can
affect the flavor evolution in a meaningful way. To gain further insight into
this question, here we study a simple system of two interacting neutrino beams,
and obtain the exact phase-space explored by this system using the Husimi
quasi-probability distribution. We observe that the entanglement induced by the
coupling leads to strong delocalization in phase-space with largely
non-Gaussian quantum fluctuations. The link between the neutrino entanglement
and quantum fluctuations is illustrated using the one- and two-neutrino
entropy. In addition, we propose an approximate phase-space method to describe
the interacting neutrinos problem, where the exact evolution is replaced by a
set of independent mean-field evolutions with a statistical sampling of the
initial conditions. The phase-space approach provides a simple and accurate
method to describe the gross features of the neutrino entanglement problem.
Applications are shown using time-independent and time-dependent Hamiltonians
in the non-adiabatic regime.
