Georgia Institute of Technology: Researchers Identify New Methods to Detect Chirality in Complex States

Shijun Sun of Georgia Institute of Technology and colleagues, working with King Fahd University of Petroleum and Minerals, have found that standard methods for identifying chirality in topological phases fail when applied to mixed quantum states. The study reveals that established diagnostics, including bulk-boundary correspondence and the modular commutator, are unreliable in these complex systems. Consequently, the research presents two new measures, based on relative entropy, to accurately diagnose chirality and determine the chiral central charge in decohered topological phases, indicating a need for a new set of tools when investigating mixed-state topology.

Quantifying decoherence via relative entropy reveals mixed-state topological properties

Relative entropy, a measure of divergence between probability distributions originating from information theory, proved central to the methodology. It quantitatively assesses the difference between a given probability distribution and a reference distribution, effectively gauging the information lost when transitioning from a pristine quantum state to a mixed, decohered one. In this context, relative entropy quantified differences between the current, mixed quantum state and a known, pristine “parent” state, assessing information loss due to decoherence. This approach circumvented the limitations of traditional diagnostics, which inherently assume a perfect, undisturbed system, an assumption rarely met in realistic materials. The mathematical formulation of relative entropy, specifically the von Neumann relative entropy, is given by S(ρ||σ) = Try(ρ log₂ ρ, ρ log₂ σ), where ρ and σ represent the density matrices of the mixed and reference states respectively. Calculating relative entropy involved considering multiple “replicas” of the quantum system, effectively creating identical copies, and analysing their collective behaviour. This replica technique amplifies subtle differences in the quantum states, enabling more sensitive detection of topological properties that might otherwise be obscured by noise. The investigation focused on mixed-state topological phases, deliberately sidestepping reliance on pristine systems, and developing new diagnostic tools applicable to decohered phases, particularly those with a known error-free starting point, providing a crucial benchmark for comparison. This focus allows for a controlled study of decoherence effects on topological properties.

Relative entropy quantifies chirality in decoherent topological phases

A team from Georgia Institute of Technology and King Fahd University of Petroleum and Minerals has demonstrated a new capability: extracting the chiral central charge, a key property defining a material’s asymmetry and directly related to the number of chiral edge modes, even when conventional methods fail. The chiral central charge, often denoted as ‘c’, is a topological invariant that characterises the strength of chiral symmetry in a system. Relative entropy-based measures successfully diagnosed chirality in these complex systems, surpassing the limitations of techniques like bulk-boundary correspondence and modular commutator analysis, both rendered ineffective by decoherence. Bulk-boundary correspondence, a cornerstone of topological physics, predicts the existence of gapless edge states arising from a gapped bulk, but this correspondence breaks down in mixed states due to the loss of coherence. Similarly, the modular commutator, a measure of non-abelian statistics, becomes ill-defined in the presence of significant decoherence. Analysis of the ZN toric code, a well-established model topological phase exhibiting anyonic excitations, showed these new measures function even with anyon condensation and decoherence, processes that typically obscure chiral signals. Anyon condensation refers to the process where these exotic particles bind together, altering the topological order. Extending these diagnostics to fully unknown mixed states remains a significant challenge, requiring the development of techniques to establish a suitable reference state without prior knowledge of the system’s origin. The current focus is on phases derived from known “parent” states, providing a controlled environment for validating the methodology. This work builds on the initial methodology by demonstrating its application to chirality, validating it using a well-established model system and providing a concrete example of its efficacy.

Strong chirality diagnostics overcome decoherence in imperfect topological materials

Diagnosing topological order in materials is already challenging, demanding precise measurements of subtle quantum properties, often requiring extremely low temperatures and high magnetic fields. A fundamental problem has been revealed: tools used to confirm chirality, a property akin to handedness at the quantum level, falter when applied to real-world materials riddled with imperfections. These imperfections introduce disorder and lead to decoherence, the loss of quantum information, effectively washing out the delicate signatures of topological order. This isn’t merely a technical hurdle; it exposes a core tension between pristine theoretical models, which often assume ideal conditions, and the messy reality of mixed quantum states, where quantum information is lost through decoherence. The abstract highlights that mathematically, chirality is well-defined through symmetry algebra, but physically realising and confirming this in a mixed state is far more complex.

Acknowledging the struggles of standard methods with imperfect materials does not diminish the importance of this development. New, strong measures of chirality, based on comparing quantum states via relative entropy, have been identified, offering a pathway to definitively characterise topological order even with real-world imperfections introducing noise. These diagnostics will be important for verifying predictions of topological physics in practical materials, enabling progress in areas like quantum computing and advanced materials’ science. Specifically, robust topological materials are crucial for building fault-tolerant quantum computers, as their inherent protection against local perturbations can safeguard quantum information. Furthermore, understanding and controlling topological phases opens doors to designing novel materials with tailored electronic and optical properties.

Assessing topological phases, materials exhibiting unusual electronic behaviours such as protected edge states and fractionalised excitations, requires new analytical techniques when dealing with real-world imperfections. Conventional methods for identifying chirality prove unreliable in these complex, disordered quantum states, losing quantum information through decoherence. Instead, diagnostic tools based on relative entropy quantify information loss when comparing a disordered state to a known, perfect one, allowing for a more sensitive detection of topological properties than previously possible. This approach offers a significant advancement in the field, bridging the gap between theoretical predictions and experimental observations in imperfect, yet potentially revolutionary, materials.

The researchers demonstrated that standard methods for identifying chirality are unreliable in mixed-state topological phases, where quantum information is lost through decoherence. They developed two new measures, based on relative entropy, to diagnose chirality in these imperfect systems and even extract the chiral central charge. This is important because it provides a way to characterise topological order in materials exhibiting unusual electronic behaviours, despite real-world imperfections introducing noise. The authors suggest these diagnostics will be crucial for verifying predictions of topological physics in practical materials.