  In a closed system, the total number of particles is fixed. We ask how much
does this conservation law restrict the amount of entanglement that can be
created. We derive a tight upper bound on the bipartite entanglement entropy in
closed systems, and find what a maximally entangled state looks like in such a
system. Finally, we illustrate numerically on an isolated system of
one-dimensional fermionic gas, that the upper bound can be reached during its
unitary evolution, when starting in a pure state that emulates a thermal state
with high enough temperature. These results are in accordance with current
experiments measuring R\'enyi-2 entanglement entropy, all of which employ a
particle-conserving Hamiltonian, where our bound acts as a loose bound, and
will become especially important for bounding the amount of entanglement that
can be spontaneously created, once a direct measurement of entanglement entropy
becomes feasible.
