Properties of Modular Hamiltonians on Entanglement Plateaux

TL;DR

This paper investigates when the modular Hamiltonian contributes non-trivially to the Fisher information metric for states forming entanglement plateaux, with implications for AdS/CFT correspondence and quantum information measures.

Contribution

It identifies conditions under which the modular Hamiltonian's contribution to the relative entropy is higher order, especially in entanglement plateau scenarios.

Findings

At least one relative entropy involves higher order modular Hamiltonian contributions.

Modular Hamiltonian contributions can be non-trivial in entanglement plateau configurations.

Implications for AdS/CFT examples in the large N limit.

Abstract

The modular Hamiltonian of reduced states, given essentially by the logarithm of the reduced density matrix, plays an important role within the AdS/CFT correspondence in view of its relation to quantum information. In particular, it is an essential ingredient for quantum information measures of distances between states, such as the relative entropy and the Fisher information metric. However, the modular Hamiltonian is known explicitly only for a few examples. For a family of states $\rho_\lambda$ that is parametrized by a scalar $\lambda$, the first order contribution in $\tilde\lambda=\lambda-\lambda_0$ of the modular Hamiltonian to the relative entropy between $\rho_\lambda$ and a reference state $\rho_{\lambda_0}$ is completely determined by the entanglement entropy, via the first law of entanglement. For several examples, e.g. for ball-shaped regions in the ground state of CFTs, higher order contributions are known to vanish. In these cases the modular Hamiltonian contributes to the Fisher information metric in a trivial way. We investigate under which conditions the modular Hamiltonian provides a non-trivial contribution to the Fisher information metric, i.e. when the contribution of the modular Hamiltonian to the relative entropy is of higher order in $\tilde{\lambda}$. We consider one-parameter families of reduced states on two entangling regions that form an entanglement plateau, i.e. the entanglement entropies of the two regions saturate the Araki-Lieb inequality. We show that in general, at least one of the relative entropies of the two entangling regions is expected to involve $\tilde{\lambda}$ contributions of higher order from the modular Hamiltonian. Furthermore, we consider the implications of this observation for prominent AdS/CFT examples that form entanglement plateaux in the large $N$ limit.