Entanglement and the sign structure of quantum states

Many-body quantum eigenstates of generic Hamiltonians at finite-energy density typically satisfy the volume law of entanglement entropy: the von Neumann entanglement entropy and the Renyi entropies for a subregion scale in proportion to its volume. Here we provide a connection between the volume law and the sign structure of eigenstates. In particular, we ask the following question: Can a positive wave function support a volume law entanglement? Remarkably, we find that a typical random positive wave function exhibits a constant law for Renyi entanglement entropies S-n for n > 1, despite arbitrary large-amplitude fluctuations. We also provide evidence that the modulus of the finite-energy density eigenstates of generic local Hamiltonians shows similar behavior.