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Topological Holography for Mixed-State Phases and Phase Transitions

Ran Luo1,2,*, Yi-Nan Wang1,3,†, and Zhen Bi4,‡

  • *Contact author: ranluo@pku.edu.cn
  • †Contact author: ynwang@pku.edu.cn
  • ‡Contact author: zjb5184@psu.edu
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PRX Quantum 6, 040358 – Published 8 December, 2025

DOI: https://doi.org/10.1103/9kmh-gjf8

Abstract

We extend the symmetry topological field theory (SymTFT) framework to open quantum systems. Using canonical purification, we embed mixed states into a doubled (2+1)-dimensional topological order and employ the slab construction to study (1+1)-dimensional mixed-state phases through condensable algebras in the doubled SymTFT. Hermiticity and positivity of the density matrix impose additional constraints on allowable 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 present examples of mixed-state phase transitions involving SWSSB and show how gauging within the open SymTFT framework reveals connections among different mixed-state phases.

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Popular Summary

Symmetry plays a central role in modern physics, shaping how materials behave and how quantum information is protected. These ideas have been especially successful in closed quantum systems, where many intriguing phases have been identified over the past decades. But real quantum devices are never isolated: they constantly interact with the environment, becoming open systems described by mixed states. This raises a fundamental question: How does symmetry behave in the presence of noise, and what kinds of quantum phases can emerge in open systems?

In this work, we generalize the symmetry topological field theory (SymTFT) framework—a powerful tool developed across high-energy, condensed matter and mathematical physics—to open quantum systems. The central idea is to view the symmetry of a d-dimensional mixed state through a holographic lens by embedding it into a (d+1)-dimensional topological system, where the symmetry structure becomes more transparent. We derive the consistency conditions required for applying this framework to mixed states and use them to determine which phases of matter can exist in the presence of noise.

With this approach, we identify both familiar pure-state phases and mixed-state phases with no analogue in pure systems—including symmetry-breaking patterns and symmetry-protected topological order that appear only in noisy environments. Beyond these conceptual advances, the SymTFT also offers practical guidance for explicit lattice constructions and low-depth circuit protocols for preparing mixed-state phases in realistic quantum devices. Altogether, our results provide a new tool for exploring how symmetry and topology interplay—and can be harnessed—in noisy quantum environments.

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PRX Quantum 6, 040358– Published 8 December, 2025
  • Received 17 August 2025
  • Revised 29 October 2025
  • Accepted 5 November 2025
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