Entropy scaling law and the quantum marginal problem

TL;DR

This paper explores the relationship between entropy scaling laws in quantum many-body states and their representation through marginal density matrices, providing a proof for a restricted case and implications for physical models.

Contribution

It proves a restricted version of the conjecture linking entropy scaling laws to the existence of a consistent global quantum state in two dimensions.

Findings

Translationally invariant marginals obeying entropy constraints are consistent with a global state.

Derived a maximum entropy density expression compatible with the marginals.

Applicable to models of topological order and certain quantum spin Hamiltonians.

Abstract

Quantum many-body states that frequently appear in physics often obey an entropy scaling law, meaning that an entanglement entropy of a subsystem can be expressed as a sum of terms that scale linearly with its volume and area, plus a correction term that is independent of its size. We conjecture that these states have an efficient dual description in terms of a set of marginal density matrices on bounded regions, obeying the same entropy scaling law locally. We prove a restricted version of this conjecture for translationally invariant systems in two spatial dimensions. Specifically, we prove that a translationally invariant marginal obeying three non-linear constraints -- all of which follow from the entropy scaling law straightforwardly -- must be consistent with some global state on an infinite lattice. Moreover, we derive a closed-form expression for the maximum entropy density compatible with those marginals, deriving a variational upper bound on the thermodynamic free energy. Our construction's main assumptions are satisfied exactly by solvable models of topological order and approximately by finite-temperature Gibbs states of certain quantum spin Hamiltonians.