Entanglement Reveals Hidden Order in Complex Materials

Researchers are increasingly focused on understanding how fragile quantum entanglement behaves in realistic, imperfect systems. Kang-Le Cai and Meng Cheng, both from the Department of Physics at Yale University, have investigated universal entanglement signatures in mixed states arising from the decoherence of topologically ordered phases. Their work centres on topological entanglement negativity and mutual information, revealing how these quantities relate to the dimensions of defects forming between different decoherence-induced boundary conditions. By developing a replica field-theory framework and applying it to decohered string-net states, the authors demonstrate a crucial distinction between topological mutual information, which probes the full emergent anyon theory, and topological entanglement negativity, which specifically detects its modular component. This research provides fundamental insights into characterising topological order in noisy quantum systems and advances our ability to detect and protect quantum information.

Within a cryostat chilled to near absolute zero, delicate quantum states are being deliberately disrupted to explore the boundaries of order. These controlled disturbances reveal hidden connections between entanglement and the underlying structure of matter. By measuring how entanglement responds to this ‘decoherence’, physicists are mapping the properties of exotic quantum phases and the anyons within them.

Scientists are increasingly focused on understanding mixed-state phases of matter, particularly those arising from the decoherence of topologically ordered systems. Topological order, a property of certain quantum materials, promises robustness against local perturbations, yet real-world materials inevitably experience noise. As a result, identifying universal characteristics within these imperfect, mixed states becomes a central challenge.

Recent work has concentrated on entanglement measures, quantifications of quantum connectedness, as potential indicators of underlying topological order even when systems are disturbed. Specifically, researchers have been examining topological entanglement negativity (TEN) and topological mutual information (TMI), which represent mixed-state analogues of the topological entanglement entropy known from pure states.

A new investigation details how to compute TEN and TMI across a broad range of decohered topological states. The approach centres on relating these entanglement measures to the dimensions of defects appearing at boundaries created by the decoherence process. By employing a replica field-theory framework, calculations can be mapped to determining the quantum dimensions of these boundary defects within a three-dimensional topological quantum field theory.

This allows for general expressions for TEN and TMI to be derived, offering a pathway to characterise mixed states. The study extends beyond purely theoretical calculations, examining decohered G-graded string-net states, complex models incorporating non-Abelian anyons. These calculations reveal a connection between TMI and the total quantum dimension of the anyon theory governing the mixed state, while TEN appears to detect only the modular portion.

Once understood, these distinctions provide a more precise method for identifying and classifying topological phases even in the presence of significant noise. Discerning the subtle signatures of topological order in noisy systems requires careful consideration of how entanglement behaves in mixed states. Unlike pure states, where entanglement entropy and mutual information directly reflect quantum correlations, mixed states also contain classical correlations.

Therefore, measures like TMI may include contributions from both quantum and classical sources. By contrast, entanglement negativity focuses more specifically on quantum entanglement, offering a cleaner signal for detecting topological order. At the core of this work lies the idea that TEN can serve as a reliable indicator of a topological phase, persisting until a certain level of decoherence is reached.

Entanglement negativity and mutual information reveal domain-wall dimensions in decohered topological phases

A replica field-theory framework, built upon a doubled-state construction, underpins the investigation of universal entanglement signatures in mixed-state phases derived from decohering pure-state topological order. This approach connects topological entanglement negativity (TEN) and topological mutual information (TMI) to the dimensions of domain-wall defects arising from decoherence-induced topological boundary conditions.

The work focuses on the strong-decoherence regime, allowing for general expressions to be derived. Once established, researchers applied these expressions to decohered -graded string-net states, encompassing scenarios with non-Abelian anyons, to refine the understanding of these entanglement measures. The methodology incorporates a strong one-form-symmetry framework for mixed-state topological order.

Researchers interpret TMI as probing the total dimension of the emergent premodular anyon theory, while TEN detects only the modular component. Detailed analysis of the representation theory of string operator algebras was performed, relating representations to those on a torus, and a concrete method for determining the dimension of irreducible representations was reviewed.

For configurations involving multiple junctions, the minimal total dimension was calculated as M m−1 n, leading to a quantum dimension of dn = √Mn and a corresponding expression for topological entanglement negativity, γEN = 1 2(n −2) ln Mn. To compute topological mutual information, the Rényi entanglement entropy was calculated by expressing the trace of the reduced density matrix as an inner product within the field-theoretic framework.

Here, γMI was extracted directly as the topological contribution to the entanglement entropy, avoiding reliance on traditional annulus geometry. By considering domain walls between specific regions, A′ 0 and A′ 2, researchers identified that decoherence does not affect A′ 2, simplifying the analysis. For a configuration with four junctions, string operators connecting the regions were defined, and the minimal-dimension representation of the associated string operator algebra yielded γMI = 1 2(n −1) ln Mn.

The framework was validated by demonstrating its ability to reproduce expected TEN and TMI values for a pure Abelian topological order, where γEN = γMI = ln D, with D representing the total quantum dimension. Beyond the Abelian case, the methodology was extended to investigate decohered topological order resulting from proliferating a subgroup of anyons generated by a single anyon, x. By analysing the Lagrangian subgroup A′ 0, researchers identified subsets based on braiding phases with x, allowing for a detailed characterisation of the decohered state.

Topological mutual information fully characterises premodular anyon theory quantum dimensions

Calculations reveal that topological mutual information (TMI) accurately quantifies the total quantum dimension of the emergent premodular anyon theory within mixed-state topological orders. Specifically, the research demonstrates TMI’s ability to probe the complete set of anyonic excitations, providing a measure directly linked to the underlying topological phase.

Conversely, topological entanglement negativity (TEN) only detects the modular portion of this anyon theory, focusing on a subset of the anyonic properties. Replica field-theory frameworks, built upon a doubled-state construction, establish a connection between TEN and TMI with the dimensions of domain-wall defects arising from decoherence-induced topological boundary conditions.

Further analysis of decohered G-graded string-net states, including those with non-Abelian anyons, yields exact computations of both TEN and TMI. These calculations confirm the theoretical framework, showing how these entanglement measures respond to decoherence. For instance, the study finds that TMI remains sensitive to the full quantum dimension even when decoherence introduces non-Abelian anyons, while TEN’s response is limited to the modular aspects.

At a strong decoherence regime, general expressions for both TEN and TMI were derived, linking them to the quantum dimensions of domain-wall defects. These findings are interpreted through the lens of strong one-form-symmetry, a key concept in understanding mixed-state topological order. Once a region’s size becomes sufficiently large, the calculations show that TMI consistently reflects the total quantum dimension, while TEN isolates the modular component.

By examining the behaviour of these measures, researchers can discern the underlying topological structure even in noisy systems. The work highlights that the logarithmic entanglement negativity, when applied to a Z2 toric code experiencing bit-flip noise, equals ln 2 when the error rate, denoted as ‘p’, is below the decoding threshold ‘pc’. Yet, when ‘p’ exceeds ‘pc’, this measure vanishes, indicating a loss of topological order.

Entanglement negativity reveals strong topological order in disordered quantum systems

Scientists are beginning to map the subtle fingerprints of order within quantum chaos, a development with implications extending beyond fundamental physics and into materials science. For years, discerning genuine topological order from mere mimicry in noisy quantum systems proved exceptionally difficult. Traditional indicators relied on pristine conditions, yet real materials invariably suffer from decoherence, the loss of quantum information to the environment.

This research offers a new set of tools for identifying these hidden topological states, even when markedly degraded by such disturbances. The power of this approach resides not simply in detecting order, but in quantifying its durability. By focusing on entanglement properties, specifically, topological entanglement negativity and mutual information, researchers have established a link between the dimensions of defects within the system and the degree of topological order remaining after decoherence.

Unlike previous methods, this framework provides a way to probe the ‘premodular’ components of the anyon theory, revealing aspects of the underlying quantum state that would otherwise be obscured. Interpreting these signals demands careful consideration. While the calculations demonstrate a clear relationship between entanglement and topological properties, applying this to complex materials presents a considerable challenge.

Determining the precise nature of the decoherence affecting a given system, and accurately modelling the resulting defect networks, will require substantial computational resources. The reliance on specific mathematical structures, namely, string-net states, limits the immediate applicability to a broader range of topological phases. Now, the field stands poised to explore mixed states with greater confidence.

Once refined, these techniques could accelerate the search for materials hosting strong topological order, potentially leading to advances in quantum computing and fault-tolerant quantum technologies. Instead of solely pursuing perfect quantum states, researchers can now investigate the possibility of leveraging imperfect, yet topologically protected, systems. Extending this framework to encompass non-Abelian anyons and more complex decoherence scenarios represents a logical next step, promising a deeper understanding of quantum matter in all its messy, real-world glory.