Mixed State Entanglement Measures in Topological Orders

We study two mixed state entanglement measures in topological orders: the so-called "computable cross-norm or realignment" (CCNR) negativity, and the more well-known partial-transpose (PT) negativity, both of which are based on separability criteria. We first compute the CCNR negativity between two spatial regions for tripartite pure states in (2+1)D Chern-Simons (CS) theories using the surgery method, and compare to the previous results on PT negativity. Under certain simplifying conditions, we find general expressions of both mixed state entanglement measures and relate them to the entanglement entropies of different subregions. Then we derive general formulas for both CCNR and PT negativities in the Pauli stabilizer formalism, which is applicable to lattice models in all spatial dimensions. Finally, we demonstrate our results in the $\mathbb{Z}_2$ toric code model. For tripartitions without trisection points, we provide a strategy of extracting the provably topological and universal terms in both entanglement measures. In the presence of trisection points, our result suggests that the subleading piece in the CCNR negativity is topological, while that for PT is not and depends on the local geometry of the trisections.