  The study of entanglement between bosonic systems is of primary importance
for establishing feasible resources needed for implementing quantum information
protocols, both in their interacting atomic or photonic realizations. Atomic
systems are particularly efficient in the production of large amounts of
entanglement, providing higher information density than conventional qubit
entangled states. Such increased quantum resources pave the way to novel
fundamental tests of nature and efficient applications in quantum information,
metrology and sensing. We consider a basic setup made up of two parties A and
B, each one populated by a single level bosonic variable. The bosons are
interacting and can hop between A and B, thus describing a two-site
Bose-Hubbard Hamiltonian. We consider the generation of quantum states in
several situations that cover the majority of physical realizations: ground
state, finite temperature, unitary dynamics, dissipation through dephasing and
loss of particles. The system is analyzed through truncated exact
diagonalization, as a function of the microscopic parameters. The non
separability of the obtained quantum states is estimated by means of the
negativity, which has recently been proven to be a suitable measure of
entanglement. Finally, we calculate lower bounds of the entanglement of
formation, an indicator that quantifies the minimal amount of entanglement
resources needed to build up such states.
