Entropy scaling law and the quantum marginal problem

• URL: http://arxiv.org/abs/2010.07424v2
• Date: Wed, 3 Feb 2021 22:56:01 GMT
• Title: Entropy scaling law and the quantum marginal problem
• Authors: Isaac H. Kim
• Abstract summary: Quantum many-body states that frequently appear in physics often obey an entropy scaling law. We prove a restricted version of this conjecture for translationally invariant systems in two spatial dimensions. We derive a closed-form expression for the maximum entropy density compatible with those marginals.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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.

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