Multipartite Entanglement in Stabilizer Tensor Networks

TL;DR

It is demonstrated that, for generic stabilizer tensor networks, the geometry of the tensor network informs the multipartite entanglement structure of the state, and implies a new operational interpretation of the monogamy of the Ryu-Takayanagi mutual information and an entropic diagnostic for higher-partiteEntanglement.

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.

Figures (3)

• Figure 1: Stabilizer tensor networks. A tensor network state is obtained by placing random stabilizer states at the bulk vertices (blue) and contracting according to the edges of the graph. In the limit of large bond dimensions, the average entanglement entropy of a boundary region $A$ is proportional to the length of a minimal cut $\gamma_A$ through the network (dashed line) hayden2016holographic, $S(A) \simeq S_{RT}(A)$, reproducing the Ryu-Takayanagi formula in holography.
• Figure 2: Tripartite entanglement and the GHZ spin model. (a) Tripartition of the boundary. (b) Illustration of the spin model (with boundary conditions and minimal energy configuration) used to evaluate the GHZ content of a random stabilizer tensor network state.
• Figure 3: Multipartite entanglement structure. (a) For any tripartition, there is only a bounded number of GHZ triples (dashed triangle) and hence the entanglement is dominated by bipartite maximal entanglement (blue lines). (b) For four (and more) parties, we can likewise extract maximally entangled pairs between any two parties (blue lines). The residual state has approximately the entropies of a perfect tensor (tetrahedron). This decomposition is in one-to-one correspondence with the extreme rays of the holographic entropy cone bao2015holographic.

Theorems & Definitions (9)

• Theorem 2: Fourpartite entanglement in random stabilizer networks
• Theorem 3
• Lemma 4
• proof
• Lemma 5
• proof
• proof : Proof of formula \ref{['eq:third']} for the third moment
• Lemma 6
• proof