Computable Cross Norm in Tensor Networks and Holography

TL;DR

This paper explores the relationship between the Computable Cross Norm (CCNR) and the reflected entropy in holography, providing explicit calculations in tensor networks and CFTs, and analyzing phase transitions and analytic continuations.

Contribution

It establishes a connection between CCNR and reflected entropy in holography, with explicit calculations and phase transition analysis in tensor networks and CFTs.

Findings

CCNR relates closely to the reflected entropy in holography.

Explicit calculations show phase transitions in tensor networks and CFTs.

Analytic continuation in holography is validated for various Re9nyi parameters.

Abstract

The Computable Cross Norm (CCNR) was recently discussed in Ref.~\cite{Yin:2022toc} as a measure of multipartite entanglement in a condensed matter context. In this short note, we point out that it is closely related to the $(2,n)$-R\'enyi reflected entropy, which has been studied in the context of AdS/CFT. We discuss the calculation of the CCNR in random tensor networks as well as holographic CFTs. The holographic dual involves a backreacted entanglement wedge cross section in a geometry sourced by R\'enyi-2 cosmic branes. We perform explicit calculations for two intervals in a hyperbolic random tensor network as well the vacuum state of a 2D holographic CFT, and analyze the occurence of a connected-to-disconnected phase transition. The example illustrates the validity of the proposal for analytic continuation in holography for arbitrary values of R\'enyi parameter $n$. We comment on a symmetry-resolved generalization of this quantity.