Replica topological order and quantum error correction

Topological phases of matter offer a promising platform for quantum computation and quantum error correction. Nevertheless, unlike its counterpart in pure states, descriptions of topological order in mixed states remain relatively underexplored. Our study gives various equivalent definitions for replica topological order in mixed states. Similar to the replica trick, our definitions also involve n copies of density matrices of the mixed state. Using our framework, we can categorize topological orders in mixed states as either quantum or classical, depending on which type of information they encode. For the case of the toric code (TC) model in the presence of decoherence, we associate for each phase a quantum channel and describe the structure of the density matrix space. We show that in the quantum-topological phase, there exists a postselection-based error correction protocol that recovers the quantum information. We accomplish this using tensor networks by identifying boundary symmetry-protected topological phases to bulk anyon condensations via a bulk-boundary correspondence. As a result, we argue that the phase boundaries of the quantum replica topological mixed states correspond to the error threshold in the toric code with postselection. We anticipate our findings to be consistent in the analytical continuation as n → 1.