Modular data for mixed-state topological order

In equilibrium, a 2+1D topological order (TO) exhibits degenerate ground states on a torus, whose transformations under the torus’s modular operations are characterized by the modular S and T matrices. This modular data has long served as a key diagnostic for distinguishing pure-state topological orders. Recently, mixed-state topological orders have attracted growing attention due to their rich phenomenology and potential relevance to quantum codes. In this work, we investigate the modular data associated with mixed-state TOs. In particular, we focus on a chiral mixed-state TO obtained by decohering a non-chiral topological state. Curiously, while the pure-state counterpart of this chiral TO lacks an exact wavefunction description (especially on the lattice), the exact density matrices of the chiral mixed-state TO can be written explicitly. We analyze their modular transformations and find that they remain consistent with those of the corresponding pure-state TO, despite the system being out of equilibrium. Moreover, the modular data is robust against perturbations of the mixed state. Finally, we discuss methods for diagnosing chirality in this mixed-state context. Our results provide examples of how classic concepts of topological order in equilibrium can be meaningfully extended to non-equilibrium quantum systems.