Area laws and efficient descriptions of quantum many-body states

Abstract

It is commonly believed that area laws for entanglement entropies imply that a\nquantum many-body state can be faithfully represented by efficient tensor\nnetwork states—a conjecture frequently stated in the context of numerical\nsimulations and analytical considerations. In this work, we show that this is\nin general not the case, except in one-dimension. We prove that the set of\nquantum many-body states that satisfy an area law for all Renyi entropies\ncontains a subspace of exponential dimension. We then show that there are\nstates satisfying area laws for all Renyi entropies but cannot be approximated\nby states with a classical description of small Kolmogorov complexity,\nincluding polynomial projected entangled pair states or states of multi-scale\nentanglement renormalisation. Not even a quantum computer with post-selection\ncan efficiently prepare all quantum states fulfilling an area law, and we show\nthat not all area law states can be eigenstates of local Hamiltonians. We also\nprove translationally and rotationally invariant instances of these results,\nand show a variation with decaying correlations using quantum error-correcting\ncodes.