Symmetry taco extends topological theory to describe open quantum systems.

Symmetry governs the organisation and classification of phases of matter, dictating how systems respond to external influences. Modern research extends the established Landau paradigm to encompass open quantum systems, those interacting with their environment, utilising symmetry topological field theory (SymTFT) as a mathematical language. While SymTFT successfully describes closed, equilibrium systems, its application to open systems presents a significant challenge, addressed here by extending the framework to accommodate mixed quantum states, representing realistic physical scenarios.

A key innovation is the ‘symmetry taco’, a novel bilayer topological order designed to encapsulate both strong and weak symmetries. Strong symmetries leave the Hamiltonian, describing the system’s energy, unchanged, while weak symmetries are preserved by the system’s dynamics but not necessarily by the Hamiltonian itself. This geometrical construction facilitates the identification of correspondences between different phases, including those that are intrinsically gapless—possessing energy levels that approach zero—and those that are gapped, possessing an energy gap separating ground and excited states.

Intrinsically gapless symmetry protected topological phases (igSPTs) are characterised by gapless edge states protected by symmetry. This research establishes a mapping between igSPTs and certain gapped symmetry protected topological phases (SPTs), offering a new perspective on their relationship, and further explores a connection between igSPTs and intrinsically average SPTs (iASPTs), broadening the scope of topological classification. Recent advances acknowledge the inevitable interaction of quantum matter with its environment, necessitating the study of mixed quantum states.

The utility of the symmetry taco extends beyond simple mapping, providing a route to systematically generate mixed-state SPTs through controlled decoherence—the loss of quantum coherence. This is significant because maintaining coherence is a major challenge in building quantum technologies, and understanding how decoherence affects topological order is crucial for developing robust quantum devices. Furthermore, the framework reveals a previously unrecognised ‘anomaly’ within these mixed quantum states, indicating the potential for novel physics and the emergence of new phases of matter. The research leverages the concept of Choi states to classify short-range correlated symmetric states, providing a foundational step towards understanding the interplay between symmetry, correlation, and topological order in mixed-state systems.

The implications extend beyond fundamental physics, offering potential avenues for designing more resilient quantum error-correcting codes. Topological order, by its nature, provides inherent protection against local perturbations, making it an attractive basis for encoding quantum information. The framework also opens new avenues for exploring dualities, anomalies, and non-equilibrium criticality, representing a shift towards a more nuanced understanding of quantum matter interacting with its environment.

The study of mixed-state quantum phases represents a significant expansion beyond conventional understandings of quantum matter, moving beyond the limitations of considering only ground states. This research centres on exploring how topological order manifests and can be characterised within these mixed states, demanding new theoretical tools and approaches. A central theme concerns the role of symmetry, encompassing both conventional symmetries and the more complex realm of non-invertible symmetries, and investigations reveal how symmetries can be broken at the quantum level, manifesting as anomalies.

Current research develops tools to identify and classify quantum phases in mixed states, employing concepts such as entanglement, Rényi entropy—a measure of entanglement—and topological invariants. Researchers investigate the properties of topological phases in mixed states, including the existence of edge states and anyonic excitations—particles exhibiting exotic exchange statistics. A particularly intriguing area focuses on SPT phases and how they can be stabilised in open systems, alongside the exploration of iASPTs.

Recent work highlights the potential of heralded noise—carefully engineered noise—for actively correcting errors and stabilising quantum phases, and researchers also investigate the possibility of intrinsic mixed-state topological order, where topological order emerges naturally within mixed states. Furthermore, the application of topological holography, alongside connections to automata and dynamical systems, provides novel perspectives on understanding these complex quantum phenomena.

This body of work demonstrates a significant expansion of understanding regarding topological order, moving beyond traditionally studied pure quantum states to encompass the more realistic realm of mixed quantum states. It actively investigates how topological properties persist, and can even emerge, within systems subject to noise and decoherence, a crucial step towards practical applications in quantum technologies. The focus shifts from identifying stable phases in idealised conditions to understanding robustness and stability in open, dissipative systems, demanding new theoretical frameworks and experimental techniques.

A central theme revolves around the development of theoretical frameworks capable of describing and classifying topological phases in mixed states. The introduction of concepts like the ‘symmetry taco’ provides a novel approach to mapping relationships between different phases and identifying new anomalies, effectively extending the established paradigm of SymTFT to accommodate mixed-state systems. This framework facilitates a systematic classification of correlated states and offers pathways for generating mixed-state SPTs through controlled decoherence, opening new avenues for research and technological innovation.

Investigations actively explore the interplay between symmetry, entanglement, and topological order, utilising advanced tools such as Rényi entropy and holographic duality. Researchers demonstrate a growing interest in non-invertible symmetries and their role in creating exotic topological phases, suggesting a deeper connection between mathematical structures and physical phenomena. The emphasis on dissipative systems highlights the importance of understanding how interactions with the environment affect the stability and properties of topological phases, demanding a more realistic and nuanced theoretical approach.

The research actively establishes a clear link between topological order and quantum error correction, suggesting that the inherent robustness of topological phases can be harnessed to protect quantum information. This connection motivates the development of new topological codes and algorithms for decoding, with a focus on enhancing their resilience to noise and decoherence. The exploration of intrinsically gapless and average SPT phases further expands the landscape of potential applications, promising new breakthroughs in quantum technology.

Future work will likely concentrate on realising these theoretical predictions in physical materials and devices, and investigating the effects of specific types of noise and decoherence on topological properties will be crucial for developing robust quantum technologies. Expanding the classification of mixed-state topological phases and exploring the connection to non-equilibrium criticality represent further avenues for research, and a key direction involves developing experimental probes to directly detect and characterise topological order in mixed states, bridging the gap between theory and experiment.