A Mechanism for the R'enyi Hierarchy of Decoherence-Induced Phase Transitions

Decoherence-induced phase transitions (DIPTs) are associated with singular changes in a quantum system's entanglement structure as the decoherence strength varies. In many cases, their critical strengths depend monotonically on the R'enyi index of the entanglement diagnostic, forming a R'enyi hierarchy. We identify a general mechanism for this hierarchy based on correlation inequalities in replicated statistical models that describe these DIPTs. We demonstrate it in mathbb{Z}₂ topological stabilizer codes subject to independent phase-flip decoherence on each qubit and in a decohered rotor model with strong U(1) symmetry. When the relevant correlations diagnose the DIPTs, these inequalities imply that the critical decoherence strength is nondecreasing with integer R'enyi index Rgeq2. We show that this mechanism also applies to a class of models with correlated decoherence.