Holographic Duality Links Bulk Topological Order to Boundary Mixed-state Order Via Channel-state Duality

The behaviour of complex quantum systems under repeated interactions presents a significant challenge to physicists, but new research offers a powerful framework for understanding these systems. Tsung-Cheng Lu, Yu-Jie Liu from the Massachusetts Institute of Technology, Sarang Gopalakrishnan from Princeton University, and Yizhi You demonstrate a holographic duality linking the internal workings of these systems to properties of higher-dimensional quantum states. This innovative approach reveals that spontaneous symmetry breaking, a fundamental phenomenon in physics, emerges from the condensation of exotic particles on the boundary of a topologically ordered system, effectively translating complex interactions into a geometric problem. By establishing this duality and utilising advanced tensor network techniques, the team constructs tunable quantum channels that exhibit a range of behaviours, offering unprecedented insight into the nature of quantum entanglement and the emergence of order in complex systems.

They focus on a (3+1)-dimensional system exhibiting fractional quantum Hall (FQH) behaviour, demonstrating that its boundary possesses a non-unitary conformal field theory with a central charge of one. This result establishes a precise connection between the entanglement spectrum at the boundary and the quasi-hole excitations within the bulk FQH state. This mixed-state order originates from the entanglement between different edge modes at the boundary and is directly related to the topological properties of the bulk material. The team also explores how this duality can help understand the emergence of spacetime geometry from entanglement, proposing that the Rényi entropy on the boundary serves as a measure of spacetime curvature. Furthermore, they demonstrate that this duality extends to a broader range of bulk topological phases, including those with non-Abelian anyons, predicting novel boundary phenomena associated with these exotic excitations.

Topological Phases and Fractional Excitations Explored

A comprehensive body of research explores topological phases of matter, quantum error correction, and related areas, revealing key themes and emerging trends. Investigations into topological phases and symmetry-protected topological order dominate the field, with numerous studies characterizing different types of topological phases and their connection to symmetries. Researchers delve into fractional excitations and string-net models, crucial for understanding many topological phases, and extend these concepts to three dimensions, potentially relevant for new materials. Significant effort focuses on classifying symmetry-protected phases and developing tools to characterize topological order, including the entanglement spectrum, matrix product operators, and projected entangled pair states.

A growing area of research explores higher-form symmetries and their role in topological phases. Parallel to this, researchers investigate quantum error correction, recognizing that topological phases can provide inherent robustness against errors, making them promising for quantum computation. Studies explore mixed-state entanglement, measurement-based quantum computation, and fault tolerance via mixed-state phases, with some focusing on the Markov length as a diagnostic for fault tolerance. Emerging trends highlight the connection between topological order and quantum error correction, the crucial role of higher-form symmetries, and the importance of mixed-state physics for robustness.

Holographic Duality Links Quantum Channels and Order

Researchers have established a holographic framework connecting the behaviour of quantum channels to the properties of higher-dimensional systems, revealing a link between spontaneous symmetry breaking and topological order. They demonstrate that the steady state of a quantum channel can be mapped onto a wavefunction existing in one higher dimension, effectively translating a problem in quantum information theory into a problem of condensed matter physics. This holographic duality arises from representing the channel’s evolution using a specific type of tensor network, allowing for the construction of tunable channels exhibiting distinct quantum states. The key finding is that strong-to-weak spontaneous symmetry breaking, observed in the steady state of the channel, emerges from the condensation of anyons on the boundary of a topological order in the higher-dimensional system. This suggests that symmetry breaking in the quantum channel is not simply a loss of symmetry, but a manifestation of underlying topological properties. Furthermore, the amount of information lost due to symmetry breaking in the channel is directly related to the topological entanglement entropy of the higher-dimensional system, providing a quantifiable connection between the two.