Sandwiched Renyi Relative Entropy in AdS/CFT

TL;DR

This paper investigates the application of sandwiched Renyi relative entropy in AdS/CFT and holographic quantum error correction, establishing its equivalence with key holographic dualities and exploring numerical insights.

Contribution

It generalizes sandwiched Renyi relative entropy to von Neumann algebras and links its equality to holographic dualities, extending previous equivalence theorems.

Findings

Equality of bulk and boundary sandwiched relative entropy relates to subregion duality.

The modular operator-based definition suits finite-dimensional holographic models.

Numerical analysis of tensor networks provides insights into entropy corrections.

Abstract

We explore the role of sandwiched Renyi relative entropy in AdS/CFT and in finite-dimensional models of holographic quantum error correction. In particular, in the context of operator algebra quantum error correction, we discuss a suitable generalization of sandwiched Renyi relative entropy over finite-dimensional von Neumann algebras. It is then shown that the equality of bulk and boundary sandwiched relative Renyi entropies is equivalent to algebraic encoding of bulk and boundary states, the Ryu-Takayanagi formula, the equality of bulk and boundary relative entropy, and subregion duality. This adds another item to an equivalence theorem between the last four items established in arxiv:1607.03901. We then discuss the sandwiched Renyi relative entropy defined in terms of modular operators, and show that this becomes the definition naturally suited to the finite-dimensional models of holographic quantum error correction. Finally, we explore some numerical calculations of sandwiched Renyi relative entropies for a simple holographic random tensor network in order to obtain a better understanding of corrections to the exact equality of bulk and boundary sandwiched relative Renyi entropy.