Multipartite Entanglement in Stabilizer Tensor Networks

TL;DR

This paper investigates the structure of multipartite entanglement in stabilizer tensor networks, revealing that tripartite entanglement is limited and providing new tools for analyzing entanglement in quantum many-body systems.

Contribution

It introduces a spin model and a new formula for the third moment of stabilizer states, advancing understanding of multipartite entanglement in tensor networks.

Findings

Tripartite entanglement in stabilizer tensor networks is scarce.

The geometry of tensor networks influences multipartite entanglement structure.

New operational interpretation of monogamy of mutual information in holography.

Abstract

Despite the fundamental importance of quantum entanglement in many-body systems, our understanding is mostly limited to bipartite situations. Indeed, even defining appropriate notions of multipartite entanglement is a significant challenge for general quantum systems. In this work, we initiate the study of multipartite entanglement in a rich, yet tractable class of quantum states called stabilizer tensor networks. We demonstrate that, for generic stabilizer tensor networks, the geometry of the tensor network informs the multipartite entanglement structure of the state. In particular, we show that the average number of Greenberger-Horne-Zeilinger (GHZ) triples that can be extracted from a stabilizer tensor network is small, implying that tripartite entanglement is scarce. This, in turn, restricts the higher-partite entanglement structure of the states. Recent research in quantum gravity found that stabilizer tensor networks reproduce important structural features of the AdS/CFT correspondence, including the Ryu-Takayanagi formula for the entanglement entropy and certain quantum error correction properties. Our results imply a new operational interpretation of the monogamy of the Ryu-Takayanagi mutual information and an entropic diagnostic for higher-partite entanglement. Our technical contributions include a spin model for evaluating the average GHZ content of stabilizer tensor networks, as well as a novel formula for the third moment of random stabilizer states, which we expect to find further applications in quantum information.