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

Abstract

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 $\text{U}(1)$ symmetry. When the relevant correlations diagnose the DIPTs, these inequalities imply that the critical decoherence strength is nondecreasing with integer R\'enyi index $R\geq2$. We show that this mechanism also applies to a class of models with correlated decoherence.