  We consider free fermion systems in arbitrary dimensions and represent the
occupation pattern of each eigenstate as a classical binary string. We find
that the Kolmogorov complexity of the string correctly captures the scaling
behavior of its entanglement entropy (EE). In particular, the
logarithmically-enhanced area law for EE in the ground state and the volume law
for EE in typical highly excited states are reproduced. Since our approach does
not require bipartitioning the system, it allows us to distinguish typical and
atypical eigenstates directly by their intrinsic complexity. We reveal that the
fraction of atypical eigenstates which do not thermalize in the free fermion
system vanishes exponentially in the thermodynamic limit. Our results
illustrate explicitly the connection between complexity and EE of individual
pure states in quantum systems.
