Entanglement in tripartitions of topological orders: a diagrammatic approach

Recent studies have demonstrated that measures of tripartite entanglement can probe data characterizing topologically ordered phases to which bipartite entanglement is insensitive. Motivated by these observations, we compute the reflected entropy and logarithmic negativity, a mixed state entanglement measure, in tripartitions of bosonic topological orders using the anyon diagrammatic formalism. We consider tripartitions in which three subregions meet at trijunctions and tetrajunctions. In the former case, we find a contribution to the negativity which distinguishes between Abelian and non-Abelian order while in the latter, we find a distinct universal contribution to the reflected entropy. Finally, we demonstrate that the negativity and reflected entropy are sensitive to the $F$-symbols for configurations in which we insert an anyon trimer, for which the Markov gap, defined as the difference between the reflected entropy and mutual information, is also found to be non-vanishing.