  In quantum statistical mechanics, it is of fundamental interest to understand
how close the bipartite entanglement entropy of eigenstates of quantum chaotic
Hamiltonians is to maximal. For random pure states in the Hilbert space, the
average entanglement entropy is known to be nearly maximal, with a deviation
that is, at most, a constant. Here we prove that, in a system that is away from
half filling and divided in two equal halves, an upper bound for the average
entanglement entropy of random pure states with a fixed particle number and
normally distributed real coefficients exhibits a deviation from the maximal
value that grows with the square root of the volume of the system. Exact
numerical results for highly excited eigenstates of a particle number
conserving quantum chaotic model indicate that the bound is saturated with
increasing system size.
