On The Entanglement Entropy For Gauge Theories (1501.02593v4)

Abstract: We propose a definition for the entanglement entropy of a gauge theory on a spatial lattice. Our definition applies to any subset of links in the lattice, and is valid for both Abelian and Non-Abelian gauge theories. For $\mathbb{Z}_N$ and $U(1)$ theories, without matter, our definition agrees with a particular case of the definition given by Casini, Huerta and Rosabal. We also argue that in general, both for Abelian and Non-Abelian theories, our definition agrees with the entanglement entropy calculated using a definition of the replica trick. Our definition, however, does not agree with some standard ways to measure entanglement, like the number of Bell pairs which can be produced by entanglement distillation.

• The paper proposes a general conceptual framework for defining entanglement entropy in both Abelian and Non-Abelian gauge theories when considered on a spatial lattice.
• The definition extends the Hilbert space to include non-gauge states, uses a tensor product structure for lattice links, and aligns with existing methods like the replica trick and CHR formulation for specific cases.
• A key finding reveals that the derived entanglement entropy does not equal the number of distillable Bell pairs due to non-local operators inherent in gauge theories.

Entanglement Entropy in Gauge Theories: An Analytical Framework

The paper, authored by Sudip Ghosh, Ronak M. Soni, and Sandip P. Trivedi, presents a conceptual framework for defining entanglement entropy in gauge theories when considered on a spatial lattice. By addressing the non-trivial issue of measuring quantum entanglement in these systems, the authors propose a general definition applicable to a variety of gauge theories, encompassing both Abelian and Non-Abelian types.

Overview of Entanglement Entropy

Entanglement entropy serves as a quantitative measure of quantum correlations within a system. It captures the intrinsic 'non-classical' behavior emerging from quantum mechanics, particularly in systems featuring gauge invariance. In gauge theories, commonly used to describe fundamental forces, the lack of an obvious tensor product decomposition of the Hilbert space complicates the computation of entanglement entropy. This is due to the non-local nature of physical excitations, such as closed loops of electric or magnetic flux, which arise due to gauge invariance.

Proposed Definition and Comparisons

The authors introduce a novel definition of entanglement entropy contingent upon the choice of a subset of lattice links, examining its validity for both Abelian and Non-Abelian gauges:

1. Hilbert Space Extension: The proposed definition extends the Hilbert space beyond gauge-invariant states, incorporating both gauge and non-gauge states, thus allowing a tensor product structure. This space is constructed by taking the tensor products of the states associated with individual lattice links.
2. Agreement with Existing Frameworks: For $\mathbb{Z}_N$ and $U(1)$ gauge theories without dynamic matter, the proposed definition aligns with extant formulations, such as that by Casini, Huerta, and Rosabal (CHR), under the "electric centre" choice. It notably provides results consistent with entanglement entropy calculated via the replica trick for both Abelian and Non-Abelian cases.
3. Replica Trick and Entanglement Entropy: The paper justifies the agreement of its entropy definition with the replica trick methodology, widely used in theoretical studies to compute quantum entanglement through path integrals extended over multiple geometrical 'sheets'.

Theoretical Implications and Subtleties

The paper identifies an essential discrepancy concerning quantification in quantum information science: the entanglement entropy derived through this method does not represent the number of Bell pairs accessible through conventional entanglement distillation or dilution processes. This anomaly is attributed to the presence of non-local operators and related excitation modes inherent in gauge theories, which impede full entanglement conversion into a form compatible with the standard quantum informational measures.

Future Directions and Open Challenges

The work paves the way for further investigations in several realms:

• Exploring the precise continuum limit of the proposed definition, to ensure that lattice artifacts do not obscure crucial physical insights,
• Extending the framework to gauge theories inclusive of matter fields, which pose additional challenges,
• Investigating the coherence of this definition in dual theories—particularly useful when exploring the landscape of gauge-gravity dualities,
• Analyzing the implications of this framework on mutual information and relative entropy, representing refined measures of quantum correlations.

In conclusion, this paper considerably progresses our understanding of entanglement entropy in gauge theories, offering a robust theoretical approach needed to elucidate the complex underpinnings of quantum correlations in fundamental physics.