Complexity of fermionic states

How much information a fermionic state contains? To address this fundamental question, we define the complexity of a particle-conserving many-fermion state as the entropy of its Fock space probability distribution, minimized over all Fock representations. The complexity characterizes the minimum computational and physical resources required to represent the state and store the information obtained from it by measurements. Alternatively, the complexity can be regarded a Fock space entanglement measure describing the intrinsic many-particle entanglement in the state. We establish universal lower bound for the complexity in terms of the single-particle correlation matrix eigenvalues and formulate a finite-size complexity scaling hypothesis. Remarkably, numerical studies on interacting lattice models suggest a general model-independent complexity hierarchy: ground states are exponentially less complex than average excited states which, in turn, are exponentially less complex than generic states in the Fock space. Our work has fundamental implications on how much information is encoded in fermionic states.