insight - Quantum Computing - # Symmetry-Resolved Entanglement Entropy

Symmetry-Resolved Entanglement Entropy in Quantum Systems with Non-Abelian Symmetries: A Mathematical Framework and Application to Spin Systems

Q: How can this framework for calculating symmetry-resolved entanglement entropy be applied to understand the behavior of topological quantum systems, which are known to possess non-abelian anyonic excitations?

This framework provides a powerful tool for studying the entanglement properties of topological quantum systems, which are characterized by non-abelian anyonic excitations. Here's how it can be applied: Characterizing Anyonic Excitations: Non-abelian anyons possess non-trivial braiding statistics, meaning that exchanging their positions can alter the system's quantum state in ways not simply described by a phase factor. The symmetry-resolved entanglement entropy, particularly its behavior under operations like partial tracing (which corresponds to tracing out a region containing anyons), can provide insights into these braiding statistics. Topological Entanglement Entropy: A key signature of topological order is the topological entanglement entropy, a constant term in the entanglement entropy that is independent of the system's size. This framework allows us to isolate the contributions to entanglement entropy coming from different symmetry sectors. By focusing on the sectors associated with anyonic excitations, we can potentially extract information about the topological entanglement entropy and hence the underlying topological order. Edge States and Entanglement: Topological systems often exhibit protected edge states, which are localized at the boundaries and carry information about the bulk topology. The entanglement between a subsystem containing edge modes and the rest of the system can be studied using this framework. This could shed light on the relationship between edge states, bulk-boundary correspondence, and the non-abelian symmetry. Symmetry-Enriched Topological Phases: This framework is particularly relevant for symmetry-enriched topological (SET) phases, where the interplay between symmetry and topology leads to richer physics. By resolving the entanglement entropy with respect to the non-abelian symmetry, we can probe the distinct SET order parameters and potentially uncover new phase transitions unique to these systems.

Q: Could there be cases where focusing solely on G-local observables unintentionally obscures certain entanglement properties accessible through non-G-local but physically relevant observables?

Yes, focusing solely on G-local observables could potentially obscure certain entanglement properties. Here's why: Hidden Entanglement: Non-G-local observables might reveal entanglement between degrees of freedom that are mixed within the symmetry sectors defined by G-local observables. In other words, states that appear unentangled from the perspective of G-local observables could possess hidden entanglement detectable only through non-G-local measurements. Symmetry-Breaking Perturbations: In realistic systems, perturbations that break the symmetry G might be unavoidable. While small perturbations might not significantly affect the G-local entanglement, they could drastically alter the entanglement structure probed by non-G-local observables. Ignoring these observables might lead to an incomplete understanding of the system's robustness against symmetry-breaking effects. Dynamical Processes: During dynamical evolution, even if the initial state is well-described by G-local entanglement, non-G-local correlations might build up over time. Restricting the analysis to G-local observables would miss this dynamically generated entanglement, which could be crucial for understanding thermalization, information scrambling, and other dynamical phenomena. Physical Relevance: In certain physical contexts, non-G-local observables might be more natural or relevant for characterizing the system's properties. For instance, in systems with emergent symmetries, the physically relevant observables might not directly correspond to the underlying microscopic symmetry group G.

Core Concepts

This paper introduces a new framework for understanding and calculating entanglement entropy in quantum systems with non-abelian symmetries, focusing on the distinction between local and symmetry-preserving (G-local) observables and their impact on entanglement entropy calculations.

Abstract

Bibliographic Information: Bianchi, E., Donà, P., & Kumar, R. (2024). Non-abelian symmetry-resolved entanglement entropy. arXiv preprint arXiv:2405.00597v2.
Research Objective: This paper aims to develop a mathematical framework for calculating symmetry-resolved entanglement entropy in quantum systems with non-abelian symmetries, a topic previously unexplored beyond specific cases like the vacuum state.
Methodology: The authors utilize tools from quantum information theory, group theory, and random matrix theory. They define subsystems operationally based on subalgebras of observables and derive exact formulas for the average and variance of entanglement entropy in systems with non-abelian symmetries. They illustrate their framework using a many-body spin system with SU(2) symmetry.
Key Findings: The research introduces the concept of "G-local" observables, which respect both locality and the system's non-abelian symmetry. They demonstrate that using G-local observables leads to a block-diagonal reduced density matrix, crucial for calculating symmetry-resolved entanglement entropy. The study derives exact formulas for the average and variance of this entropy for random states within fixed symmetry sectors. Applying this to a spin system with SU(2) symmetry, they show that the entanglement entropy becomes asymmetric under subsystem exchange, a phenomenon not observed in systems with abelian symmetries.
Main Conclusions: The paper provides a rigorous framework for studying entanglement in the presence of non-abelian symmetries, highlighting the importance of G-local observables. Their findings have significant implications for understanding quantum many-body systems, particularly in the context of symmetry-protected phases and quantum information processing.
Significance: This work significantly advances the understanding of entanglement in quantum systems with non-abelian symmetries. It provides a new perspective on the interplay between symmetry, locality, and entanglement, opening avenues for exploring novel quantum phases of matter and developing new quantum technologies.
Limitations and Future Research: The paper focuses on systems with finite-dimensional Hilbert spaces. Extending this framework to infinite-dimensional systems relevant to quantum field theory would be a natural next step. Further research could explore the implications of these findings for specific physical systems and quantum information tasks.

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Quotes

"For abelian symmetries, such as number conservation or charge conservation, the notion of typical entanglement entropy [4–12] and its relation to symmetry-resolved entanglement [13–15] is well studied."
"The generalization to a non-abelian symmetry group is not immediate and requires new tools, which we introduce in this paper."
"In this paper, we study the interplay between locality, symmetries and entanglement. In particular, we show that the Page curve for the typical entanglement entropy [1, 2] captures new phenomena proper of systems with a non-abelian symmetry group [3]."
"The operational definition of a subsystem in terms of the subalgebra of G-local observables guarantees that the subsystem is both local and G-invariant."

Key Insights Distilled From

Non-abelian symmetry-resolved entanglement entropy

by Eugenio Bian... at arxiv.org 11-06-2024

https://arxiv.org/pdf/2405.00597.pdf

Non-abelian symmetry-resolved entanglement entropy

Deeper Inquiries

How can this framework for calculating symmetry-resolved entanglement entropy be applied to understand the behavior of topological quantum systems, which are known to possess non-abelian anyonic excitations?

This framework provides a powerful tool for studying the entanglement properties of topological quantum systems, which are characterized by non-abelian anyonic excitations. Here's how it can be applied:

Characterizing Anyonic Excitations: Non-abelian anyons possess non-trivial braiding statistics, meaning that exchanging their positions can alter the system's quantum state in ways not simply described by a phase factor. The symmetry-resolved entanglement entropy, particularly its behavior under operations like partial tracing (which corresponds to tracing out a region containing anyons), can provide insights into these braiding statistics.

Topological Entanglement Entropy:  A key signature of topological order is the topological entanglement entropy, a constant term in the entanglement entropy that is independent of the system's size. This framework allows us to isolate the contributions to entanglement entropy coming from different symmetry sectors. By focusing on the sectors associated with anyonic excitations, we can potentially extract information about the topological entanglement entropy and hence the underlying topological order.

Edge States and Entanglement: Topological systems often exhibit protected edge states, which are localized at the boundaries and carry information about the bulk topology. The entanglement between a subsystem containing edge modes and the rest of the system can be studied using this framework. This could shed light on the relationship between edge states, bulk-boundary correspondence, and the non-abelian symmetry.

Symmetry-Enriched Topological Phases:  This framework is particularly relevant for symmetry-enriched topological (SET) phases, where the interplay between symmetry and topology leads to richer physics. By resolving the entanglement entropy with respect to the non-abelian symmetry, we can probe the distinct SET order parameters and potentially uncover new phase transitions unique to these systems.

Could there be cases where focusing solely on G-local observables unintentionally obscures certain entanglement properties accessible through non-G-local but physically relevant observables?

Yes, focusing solely on G-local observables could potentially obscure certain entanglement properties. Here's why:

Hidden Entanglement: Non-G-local observables might reveal entanglement between degrees of freedom that are mixed within the symmetry sectors defined by G-local observables. In other words, states that appear unentangled from the perspective of G-local observables could possess hidden entanglement detectable only through non-G-local measurements.

Symmetry-Breaking Perturbations:  In realistic systems, perturbations that break the symmetry G might be unavoidable. While small perturbations might not significantly affect the G-local entanglement, they could drastically alter the entanglement structure probed by non-G-local observables. Ignoring these observables might lead to an incomplete understanding of the system's robustness against symmetry-breaking effects.

Dynamical Processes:  During dynamical evolution, even if the initial state is well-described by G-local entanglement, non-G-local correlations might build up over time.  Restricting the analysis to G-local observables would miss this dynamically generated entanglement, which could be crucial for understanding thermalization, information scrambling, and other dynamical phenomena.

Physical Relevance:  In certain physical contexts, non-G-local observables might be more natural or relevant for characterizing the system's properties. For instance, in systems with emergent symmetries, the physically relevant observables might not directly correspond to the underlying microscopic symmetry group G.

If we consider the universe as a closed quantum system with potentially unknown symmetries, how might this new understanding of entanglement entropy influence our interpretation of the second law of thermodynamics and the arrow of time?

The idea of the universe as a closed quantum system with potentially unknown symmetries, combined with this new understanding of entanglement entropy, raises intriguing questions about the second law of thermodynamics and the arrow of time:

Generalized Second Law:  If the universe possesses unknown symmetries, our current formulation of the second law, based on thermodynamic entropy, might be incomplete. A generalized second law might need to incorporate the entanglement entropy associated with these symmetries. This could potentially explain the perceived entropy increase in the universe even if the fundamental laws are time-reversal invariant.

Entanglement and Thermalization:  The process of thermalization, where systems approach equilibrium and experience entropy increase, could be influenced by the entanglement structure dictated by these unknown symmetries. Understanding how entanglement spreads through different symmetry sectors might provide insights into the mechanisms driving the arrow of time.

Cosmology and Early Universe:  In the very early universe, where quantum effects are believed to have played a significant role, the entanglement entropy associated with these unknown symmetries could have had profound implications for cosmological evolution. It might have influenced structure formation, the expansion rate, and even the initial conditions of the universe.

Fundamental Limits on Information:  The existence of unknown symmetries and their associated entanglement entropy could impose fundamental limits on our ability to acquire information about the universe.  Observables that are not invariant under these symmetries might be fundamentally inaccessible, leading to inherent uncertainties in our measurements and predictions.

Quantum Gravity and Black Holes:  Understanding the role of entanglement entropy in the presence of unknown symmetries could be crucial for developing a consistent theory of quantum gravity. It might provide insights into the information paradox of black holes, where the entanglement entropy of Hawking radiation seems to violate unitarity.