Critical Information Phase: Decoding Z_N Toric Codes under Decoherence

In this work, we study the mixed state phases that arise in Z_N Toric codes subject to decoherence. In contrast to the Z₂ Toric code, we find that for N>4, the Z_N Toric code can exhibit three distinct phases: (i) a long-range-entangled, decodable phase, (ii) a short-range-entangled, non-decodable phase, and (iii) an intermediate critical information phase. We characterize this new critical phase by computing several information-theoretic diagnostics, which we express in terms of partition functions of two-dimensional disordered clock models along the Nishimori line. The critical information phase of the decohered Z_N Toric code corresponds to the quasi-long-range-ordered (QLRO) phase of these disordered clock models. We find that the Markov length diverges in the critical-information phase, while both the coherent information and the topological entanglement negativity saturate to finite values in the thermodynamic limit, interpolating between their values in the ordered and disordered phases. This behavior reveals the persistence of fractional recoverable logical information and fractional topological order, extending the landscape of phases relevant to topological quantum memories beyond the conventional gapped paradigm. Finally, we introduce a decoding algorithm for the Z_N​ toric code based on a minimum-cost integer-flow solver followed by a Monte Carlo decoder. We demonstrate that its threshold aligns with the intrinsic decodable–critical phase transition point.