Quantum Entanglement of Local Operators in Conformal Field Theories (1401.0539v1)

Abstract: We introduce a series of quantities which characterizes a given local operator in conformal field theories from the viewpoint of quantum entanglement. It is defined by the increased amount of (Renyi) entanglement entropy at late time for an excited state defined by acting the local operator on the vacuum. We consider a conformal field theory on an infinite space and take the subsystem in the definition of the entanglement entropy to be its half. We calculate these quantities for a free massless scalar field theory in 2, 4 and 6 dimensions. We find that these results are interpreted in terms of quantum entanglement of finite number states, including EPR states. They agree with a heuristic picture of propagations of entangled particles.

• The paper demonstrates that in higher dimensions, the late-time Renyi entropies of locally excited states converge to finite, dimension-independent constants.
• It employs a decomposition of the scalar field into left- and right-moving modes, clarifying how local operators generate entangled particle pairs.
• The analysis reveals that in two dimensions, entanglement increases when an entangled pair reaches a subsystem boundary, marking a contrast with higher-dimensional behaviors.

Overview of Quantum Entanglement of Local Operators in Conformal Field Theories

The paper "Quantum Entanglement of Local Operators in Conformal Field Theories" by Nozaki, Numasawa, and Takayanagi offers a systematic investigation into the quantum entanglement properties of local operators within the framework of conformal field theories (CFTs). It presents a series of quantities designed to characterize entanglement in locally excited states defined by the action of local operators on the vacuum. The paper spans across dimensions 2, 4, and 6, utilizing the Renyi entanglement entropy as a key tool for analysis.

Key Findings

• Entanglement in High Dimensions: The research employs a free massless scalar field theory to compute late-time values of the Renyi entropic quantities for these states in multiple dimensions. The paper finds that in dimensions higher than two, the entanglement at late times approaches finite constants which are largely independent of spatial dimensions. For instance, the paper identified that at late time, these entanglement values are invariant concerning rotation in space for dimensions greater than two, with prominent examples and dimensional results summarized in Table I.
• Decomposition of Field Operators: The authors demonstrated that the decomposition of the scalar field into left-moving and right-moving modes provides insight into the formation and propagation of entangled pairs of particles. This decomposition plays a significant role in elucidating the Renyi entropy's behavior and highlights a novel interpretation of quantum entanglement arising from such local operators.
• Behavior in Two Dimensions: In lower dimensions, particularly in two dimensions, the operators considered cannot be classified as local in the conventional sense due to their vanishing conformal dimensions. However, by selecting specific primary operators, the paper demonstrates that the increase in entanglement entropy commences when an entangled pair reaches the boundary of a subsystem and becomes governed by relativistic propagation—a marked divergence from behaviors observed in higher dimensions.

Analytical Approach

The authors utilize an analytical continuation from Euclidean to Lorentzian times to explore the time evolution of the entanglement metrics. The path integral formalism enabled the incorporation of the replica method, which provides a robust basis for analysis within CFTs on infinitely large spaces.

Implications and Future Directions

• Theoretical Impact: The paper's approach offers a deeper understanding of the quantum entanglement characteristics inherent to local operators in CFTs. This can potentially refine theoretical models concerning operator entanglement in quantum field theory and provide new avenues for investigating the link between spacetime geometries and entropic properties.
• Practical Applications: While not explicitly focused on practical applications, understanding operator entanglement in this context can have implications in areas such as quantum computing and quantum information, particularly regarding efficient computation within large-scale quantum systems.
• Future Research: The paper suggests that these results could be further extended to interacting CFTs and massive quantum field theories, which could reveal complex entanglement behaviors substantial for emerging quantum technologies. Additionally, exploring holographic computations in systems with quasiparticle interpretations may yield new insights into the entanglement structure within diverse physical systems.

Overall, the paper introduces a pioneering perspective on entanglement entropy concerning local operators, providing meaningful contributions to the theoretical landscape of quantum field theories, and inciting further explorations within the domain.