Universal tripartite entanglement in one-dimensional many-body systems

Motivated by conjectures in holography relating the entanglement of purification and reflected entropy to the entanglement wedge cross-section, we introduce two related non-negative measures of tripartite entanglement $g$ and $h$. We prove structure theorems which show that states with nonzero $g$ or $h$ have nontrivial tripartite entanglement. We then establish that in 1D these tripartite entanglement measures are universal quantities that depend only on the emergent low-energy theory. For a gapped system, we argue that either $g\neq 0$ and $h=0$ or $g=h=0$, depending on whether the ground state has long-range order. For a critical system, we develop a numerical algorithm for computing $g$ and $h$ from a lattice model. We compute $g$ and $h$ for various CFTs and show that $h$ depends only on the central charge whereas $g$ depends on the whole operator content.