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We consider free fermion systems in arbitrary dimensions and represent theoccupation pattern of each eigenstate as a classical binary string. We findthat the Kolmogorov complexity of the string correctly captures the scalingbehavior of its entanglement entropy (EE). In particular, thelogarithmically-enhanced area law for EE in the ground state and the volume lawfor EE in typical highly excited states are reproduced. Since our approach doesnot require bipartitioning the system, it allows us to distinguish typical andatypical eigenstates directly by their intrinsic complexity. We reveal that thefraction of atypical eigenstates which do not thermalize in the free fermionsystem vanishes exponentially in the thermodynamic limit. Our resultsillustrate explicitly the connection between complexity and EE of individualpure states in quantum systems.

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