Multi-entropy from Linking in Chern-Simons Theory

TL;DR

This work analyzes multipartite entanglement in link states produced by Euclidean Chern-Simons path integrals for Abelian and non-Abelian gauge groups. It derives a closed-form expression for the tripartite Rényi multi-entropy in the Abelian three-component case and shows that the genuine multi-entropy captures GHZ-type tripartite entanglement in stabilizer link states. The authors extend the analysis to N- and four-component Abelian links via gcd- and kernel-structure from linking data and compute non-Abelian cases for several three-component links, linking entanglement measures to quantum dimensions and link invariants. They also relate genuine tri-entropy and logarithmic negativity to stabilizer-state decompositions, derive bounds, and discuss implications for topological phases and potential holographic connections.

Abstract

We study the multipartite entanglement structure of quantum states prepared by the Euclidean path integral over three-manifolds with multiple torus boundaries (the so-called link states) in both Abelian and non-Abelian Chern-Simons theories. For three-component link states in the Abelian theory, we derive an explicit formula for the Rényi multi-entropy in terms of linking numbers. We further show that the genuine multi-entropy faithfully quantifies the tripartite entanglement generated by GHZ-states, consistent with the fact that the prepared states are stabilizer states.

Figures (10)

• Figure 1: Graphical notation for the tripartite wavefunction $\psi$, its conjugate $\psi^*$ and the reduced density matrix $\rho_{BC}$.
• Figure 2: The construction of $\mathcal{Z}^{(\mathtt{q})}_{n^{\mathtt{q} - 1}}$ in the special case $\mathtt{q} = 3$ and $n = 2,3$.
• Figure 3: Graphical notation for the quadripartite wavefunction $\psi$, its conjugate $\psi^*$, and the construction of $\mathcal{Z}^{(\mathtt{q = 4})}_{2^{\mathtt{4} - 1}}$. Note that for clarity we omitted some identifications.
• Figure 4: Figure showing the construction of link $6_3^3$ complement. Removing a tubular neighborhood around link $6_3^3$ embedded in $\mathbb{S}^3$ results in link $6_3^3$ complement which is a manifold with three torus boundaries. We note that the color in this figure (and Fig. \ref{['fig-three-links']}) is used to distinguish different components and does not denote specific representation.
• Figure 5: (a) The connected sum of two Hopf links (the link $2_1^2 + 2_1^2$). (b) The link $6_3^3$. (c) Borromean rings (the link $6_2^3$).