Nonadditivity of Renyi entropy and Dvoretzky's theorem

We examine whether it is possible for one-dimensional translationally invariant Hamiltonians to have ground states with a high degree of entanglement. We present a family of translationally invariant Hamiltonians {H-n} for the infinite chain. The spectral gap of H-n is (1/poly(n)). Moreover, for any state in the ground space of H-n and any m, there are regions of size m with entanglement entropy (min{m,n}). A similar construction yields translationally invariant Hamiltonians for finite chains that have unique ground states exhibiting high entanglement. The area law proven by Hastings ["An area law for one dimensional quantum systems," J. Stat. Mech.: Theory Exp. 2007 (08024)] gives a constant upper bound on the entanglement entropy for one-dimensional ground states that is independent of the size of the region but exponentially dependent on 1/Delta, where Delta is the spectral gap. This paper provides a lower bound, showing a family of Hamiltonians for which the entanglement entropy scales polynomially with 1/Delta. Previously, the best known such bound was logarithmic in 1/Delta.