Topological Holography for Mixed-State Phases and Phase Transitions

Classifying phases and phase transitions of mixed quantum states is an emerging goal in many-body physics. We generalize the symmetry topological field theory (SymTFT) framework to open quantum systems. Using canonical purification, a mixed state is embedded into a doubled (2+1)D topological order; a slab construction then maps (1+1)D mixed-state phases to condensable algebras in the doubled SymTFT. Enforcing Hermiticity and positivity of the density matrix imposes new constraints on admissible anyon condensations, enabling a systematic classification of mixed-state phases, including strong-to-weak symmetry breaking (SWSSB) and average symmetry-protected topological (ASPT) phases. We also present examples of transitions out of SWSSB phase and show how gauging operations within the open SymTFT generate duality between distinct mixed-state phases. The framework offers a unifying language for understanding phases and phase transitions in open quantum systems.