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

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. To the best of our knowledge, this is the first nontrivial solution to the quantum marginal problem in a many-body setting that lies strictly outside the framework of mean-field theory.

HEP - theorymathematical physicsquantum physics

Audience: researchers in the topic

Purdue HET

Series comments: The recorded talks will be available on YouTube here: www.youtube.com/playlist?list=PLxU3vHZccQj64m9zsQR74D5WP1z1g4t1J