Computable Cross Norm in Tensor Networks and Holography

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 $(2,n)$-Rényi 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 Rényi-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 Rényi parameter $n$. We comment on a symmetry-resolved generalization of this quantity.

14 pages, 4 figures, added clarity on various points in v2