  We theoretically explore quantum correlation properties of a dissipative
Bose-Hubbard dimer in presence of a coherent drive. In particular, we focus on
the regime where the semiclassical theory predicts a bifurcation with a
spontaneous spatial symmetry breaking. The critical behavior in a well defined
thermodynamical limit of large excitation numbers is considered and analyzed
within a Gaussian approach. The case of a finite boson density is also examined
by numerically integrating the Lindblad master equation for the density matrix.
We predict the critical behavior around the bifurcation points accompanied with
large quantum correlations of the mixed steady-state, in particular exhibiting
a peak in the logarithmic entanglement negativity.
