  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.
