Area laws and efficient descriptions of quantum many-body states

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. 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 or states of multi-scale entanglement renormalisation. 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 and rotationally invariant instances of these results, and show a variation with decaying correlations using quantum error-correcting codes.

Finite Correlation Length Scaling in Lorentz-Invariant Gapless iPEPS Wave Functions

“…Coming back to the reason for the finite ξ τ (D) in the first place: We speculate that finding the dominant eigenvector of a plane-to-plane transfer operator along the temporal direction combined with a projection to a finite D iPEPS leads invariably to a finite correlation length in the temporal direction. This view is also supported by a more formal argument stating that an entropic area law does not automatically imply an efficient iPEPS representation [69].…”

Finite Correlation Length Scaling with Infinite Projected Entangled-Pair States

“…The finite D always induces a finite correlation length ξ D in the iPEPS state, in complete analogy to the 1D case. Lorentz invariant critical points could thus describe a class of states which, despite fulfilling the area law of entanglement, cannot be faithfully represented by an iPEPS with a finite D (see also [38]). As a positive consequence, this allows us to the apply the ideas of FCLS also in 2D for the accurate and systematic study of quantum critical phenomena.…”

Time evolution of an infinite projected entangled pair state: An algorithm from first principles

“…The fraction is usually set at 10 −8 . To this end we solve the linear equation (13) using the Moore-Penrose pseudo-inversẽ…”