Entanglement and ground states of gapped Hamiltonians

TL;DR

This paper derives bounds for entanglement in finitely correlated states and ground states of gapped Hamiltonians, providing new analytical tools and extending results to higher dimensions, with implications for quantum information theory.

Contribution

It introduces new bounds for entanglement in finitely correlated states and extends Hastings' ground state approximation to higher dimensions, advancing understanding of entanglement and spectral gaps.

Findings

Bounds for entanglement in finitely correlated states derived

Entropy of ground state restrictions is uniformly bounded

Multiplicativity of the output 2-norm holds for certain quantum channels

Abstract

We begin by deriving bounds for the entanglement of a spin with an (adjacent and non-adjacent) interval of spins in an arbitrary pure Finitely Correlated States (FCS). The bounds we derive become exact in the case where one considers the entanglement of a single spin with a half-infinite chain to the right of it. Our result permits a more efficient calculation, numerically and in some cases even analytically, of the entanglement in arbitrary finitely correlated quantum spin chains. We continue the study of entanglement in the setting of ground states of Hamiltonians with a spectral gap. In particular, for $V$ a finite subset of $Z^d$, we let $H_V$ denote a Hamiltonian on $V$ with finite range, finite strength interactions and a unique ground state with a non-vanishing spectral gap. For a density matrix $\rho_A$ that describes the finite-volume restriction to a region $A$ of the unique ground state, we provide a detailed version of a proof by M. Hastings, that the entropy of $\rho_A$ is bounded by a uniform constant $C$. Moreover, we provide a detailed generalization of the 1-dimensional construction of Hastings' approximation to the ground state in dimensions 2 and higher. Finally, we turn our attention to the study of a conjecture central to Quantum Information Theory, the multiplicativity of the maximal output Schatten $p$-norm of quantum channels, for $p > 1$. In particular, we study the output 2-norm for a special class of quantum channels, the depolarized Werner-Holevo channels, and show that multiplicativity holds for a product of two identical channels in this class.