  In atomically thin transition metal dichalcogenides (TMDs), reduced
dielectric screening of the Coulomb interaction leads to strongly correlated
many-body states, including excitons and trions, that dominate the optical
properties. Higher-order states, such as bound biexcitons, are possible but are
difficult to identify unambiguously using linear optical spectroscopy methods
alone. Here, we implement polarization-resolved two-dimensional coherent
spectroscopy to unravel the complex optical response of monolayer MoSe$_2$ and
identify multiple higher-order correlated states. Decisive signatures of
neutral and charged inter-valley biexcitons appear in cross-polarized
two-dimensional spectra as distinct resonances with respective ~20 meV and ~5
meV binding energies--similar to recent calculations using variational and
Monte Carlo methods. A theoretical model taking into account the
valley-dependent optical selection rules reveals the specific quantum pathways
that give rise to these states. Inter-valley biexcitons identified here,
comprised of neutral and charged excitons from different valleys, offer new
opportunities for creating exotic exciton-polariton condensates and for
developing ultrathin biexciton lasers and polarization-entangled photon
sources.
