  In a seminal paper [D. N. Page, Phys. Rev. Lett. 71, 1291 (1993)], Page
proved that the average entanglement entropy of subsystems of random pure
states is $S_{\rm ave}\simeq\ln{\cal D}_{\rm A} - (1/2) {\cal D}_{\rm
A}^2/{\cal D}$ for $1\ll{\cal D}_{\rm A}\leq\sqrt{\cal D}$, where ${\cal
D}_{\rm A}$ and ${\cal D}$ are the Hilbert space dimensions of the subsystem
and the system, respectively. Hence, typical pure states are (nearly) maximally
entangled. We develop tools to compute the average entanglement entropy
$\langle S\rangle$ of all eigenstates of quadratic fermionic Hamiltonians. In
particular, we derive exact bounds for the most general translationally
invariant models $\ln{\cal D}_{\rm A} - (\ln{\cal D}_{\rm A})^2/\ln{\cal D}
\leq \langle S \rangle \leq \ln{\cal D}_{\rm A} - [1/(2\ln2)] (\ln{\cal D}_{\rm
A})^2/\ln{\cal D}$. Consequently we prove that: (i) if the subsystem size is a
finite fraction of the system size then $\langle S\rangle<\ln{\cal D}_{\rm A}$
in the thermodynamic limit, i.e., the average over eigenstates of the
Hamiltonian departs from the result for typical pure states, and (ii) in the
limit in which the subsystem size is a vanishing fraction of the system size,
the average entanglement entropy is maximal, i.e., typical eigenstates of such
Hamiltonians exhibit eigenstate thermalization.
