The modern theory of phases of matter centers on gapped ground states and their remarkable locality properties. Extending these ideas to mixed states remains challenging, as many physically relevant states (e.g., thermal states, steady states of open systems, and metastable states) are not naturally described by a gapped local parent Hamiltonian. Moreover, mixed states can exhibit correlations that are invisible to ordinary two-point functions, suggesting that their correlation structure is inherently richer. Hence, non-equilibrium states can exhibit forms of criticality that extend beyond the conventional framework of long-range order, suggesting that new principles may be needed to characterize phases in mixed-state settings.

In this talk, I will propose a characterization of gapped mixed states through an extended notion of short-range correlations (SRC), defined by the decay of both ordinary correlations and conditional mutual information. I will show that this condition is equivalent to a robust operational property, termed local stability, stating that local operations can have only local effects and can be reversed by local channels. I will also discuss several consequences of this framework, including the decay of a broad class of nonlinear connected correlation functions, thereby ruling out critical phenomena such as long-range order and Strong-weak SSB. Finally, we demonstrate the decay of correlations in the canonical purifications of such SRC states, relating mixed-state phases to pure-state phases.

Taken together, these results suggest that our definition provides a unified notion of phase applicable to both pure and mixed states.