Entropy Scaling Law and the Quantum Marginal Problem

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