Area laws and efficient descriptions of quantum many-body states

TL;DR

This paper challenges the common belief that area laws for entanglement entropies guarantee efficient tensor network representations of quantum many-body states, showing this is only true in one dimension and providing counterexamples in higher dimensions.

Contribution

It proves that area law states in higher dimensions cannot always be efficiently approximated by classical or tensor network states, linking quantum many-body theory with communication complexity.

Findings

Area law states in higher dimensions form a large, complex set.

Not all area law states can be efficiently prepared by quantum computers.

Some area law states are not eigenstates of local Hamiltonians.

Abstract

It is commonly believed that area laws for entanglement entropies imply that a quantum many-body state can be faithfully represented by efficient tensor network states - a conjecture frequently stated in the context of numerical simulations and analytical considerations. In this work, we show that this is in general not the case, except in one dimension. We prove that the set of quantum many-body states that satisfy an area law for all Renyi entropies contains a subspace of exponential dimension. Establishing a novel link between quantum many-body theory and the theory of communication complexity, we then show that there are states satisfying area laws for all Renyi entropies but cannot be approximated by states with a classical description of small Kolmogorov complexity, including polynomial projected entangled pair states (PEPS) or states of multi-scale entanglement renormalisation (MERA). Not even a quantum computer with post-selection can efficiently prepare all quantum states fulfilling an area law, and we show that not all area law states can be eigenstates of local Hamiltonians. We also prove translationally invariant and isotropic instances of these results, and show a variation with decaying correlations using quantum error-correcting codes.