  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.
