# Multipartite entanglement and quantum Fisher information in conformal field theories

## TL;DR

This paper explores how quantum Fisher information relates to multipartite entanglement in 1+1 dimensional conformal field theories, linking entanglement properties to central charge and scaling dimensions, and analyzing their behavior under renormalization group flow.

## Contribution

It introduces a novel connection between quantum Fisher information and multipartite entanglement in CFTs, emphasizing the role of the smallest scaling dimension and RG flow effects.

## Key findings

- Larger central charge correlates with more multipartite entanglement.
- Smaller smallest scaling dimension leads to higher quantum Fisher information.
- Quantum Fisher information decreases along renormalization group flow.

## Abstract

Bipartite entanglement entropy of a segment with the length $l$ in $1+1$ dimensional conformal field theories (CFT) follows the formula $S=\frac{c}{3}\ln l+\gamma$, where $c$ is the central charge of the CFT and $\gamma$ is a cut-off dependent constant which diverges in the absence of an ultraviolet cutoff. According to this formula, systems with larger central charges have {\it{more}} bipartite entanglement entropy. Using quantum Fisher information (QFI), we argue that systems with bigger central charges not only have larger bipartite entanglement entropy but also have more multipartite entanglement content. In particular, we argue that since systems with smaller {\it{smallest scaling dimension}} have bigger QFI, the multipartite entanglement content of a CFT is dependent on the value of the smallest scaling dimension present in the spectrum of the system. We show that our argument seems to be consistent with some of the existing results regarding the von Neumann entropy, negativity, and localizable entanglement in $1+1$ dimensions. Furthermore, we also argue that the QFI decays under renormalization group (RG) flow between two unitary CFTs. Finally, we also comment on the non-conformal but scale invariant systems.