Kolmogorov complexity as intrinsic entropy of a pure state: Perspective from entanglement in free fermion systems

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.