Entropy scaling law and the quantum marginal problem

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.

61 pages, 7 figures, ~200 in-line figures. Added the toric code example and a discussion about an efficient tensor network construction. More detailed explanation added in Section II

References in corpus (5)

• Lieb-Robinson bounds and the generation of correlations and topological quantum order
• Ground state entanglement and geometric entropy in the Kitaev's model
• Entanglement and topological entropy of the toric code at finite temperature
• Approximating Gibbs states of local Hamiltonians efficiently with PEPS
• Entanglement bootstrap approach for gapped domain walls

Cited by in corpus (6)

• Universal lower bound on topological entanglement entropy
• Many-body entropies and entanglement from polynomially-many local measurements
• Nonlocal growth of quantum conditional mutual information under decoherence
• Clustering of conditional mutual information and quantum Markov structure at arbitrary temperatures
• Stacked tree construction for free-fermion projected entangled pair states
• An exactly solvable ansatz for statistical mechanics models