Entanglement and the Sign Structure of Quantum States

TL;DR

This paper explores the relationship between entanglement entropy and the sign structure of quantum states, revealing that positive wavefunctions generally do not support volume law entanglement, unlike typical eigenstates.

Contribution

It demonstrates that positive wavefunctions typically exhibit only constant Renyi entanglement entropy, challenging assumptions about volume law entanglement in such states.

Findings

Random positive wavefunctions have constant Renyi entropies for n>1.

Finite energy eigenstates of local Hamiltonians show similar entanglement behavior.

Sign structure influences the entanglement scaling in quantum states.

Abstract

Many body quantum eigenstates of generic Hamiltonians at finite energy density typically satisfy "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 question: can a positive wavefunction support a volume law entanglement? Remarkably, we find that a typical random positive wavefunction, 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 show similar behavior.