Entropy scaling law and the quantum (and classical) marginal problem

Speaker: Isaac Kim
Affiliation: University of Sydney

Abstract

Quantum and classical many-body states that 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. This talk will focus on the explanation of these results for classical systems, which is significantly simpler than their quantum counterparts. Towards the end, I will also briefly discuss how a similar idea can be applied in a quantum setting.

This talk is based on arXiv:2010.07423 and arXiv:2010.07424.