Entanglement entropy in free quantum field theory (0905.2562v3)

Abstract: In this review we first introduce the general methods to calculate the entanglement entropy for free fields, within the Euclidean and the real time formalisms. Then we describe the particular examples which have been worked out explicitly in two, three and more dimensions.

• The paper establishes a robust framework by combining Euclidean and real-time approaches to calculate entanglement entropy in free quantum field theories.
• The study demonstrates that the leading entropy scales with the area of the boundary, with divergences arising from the short-distance cutoff.
• Numerical and analytical examples reveal universal logarithmic corrections that have significant implications for quantum computing, black hole thermodynamics, and condensed matter physics.

Entanglement Entropy in Free Quantum Field Theory

The paper "Entanglement entropy in free quantum field theory" authored by H. Casini and M. Huerta, explores the rich framework of quantum entanglement entropy within the field of free quantum field theories (QFT). A pivotal quantity in quantum mechanics and quantum information theory, entanglement entropy is pertinent across various domains such as quantum computing, black hole thermodynamics, and condensed matter physics.

Overview

The document discusses methodologies to compute the entanglement entropy for free quantum fields by leveraging both Euclidean and real-time field theory formalisms. This task primarily involves addressing the computational challenges associated with tracing over degrees of freedom, especially when dealing with bosonic and fermionic fields. The theoretical exploration is supported by specific examples illustrating these calculations in various dimensions.

Key Concepts and Results

1. Entanglement Entropy Definition:
   • The entanglement entropy $S(V)$ for a spatial region $V$ in QFT is calculated from the reduced density matrix $\rho_V$, derived by tracing over the degrees of freedom outside $V$. The entropy is given by the von Neumann formula: $S(V) = -\text{tr}(\rho_V \log \rho_V)$.
2. Area Law and Divergences:
   • In any QFT, the leading term in the entanglement entropy scales with the area of the boundary, known as the area law. This leads to divergences categorized by degrees of the short-distance cutoff $\epsilon$ in the entropy formula.
3. Euclidean Approach:
   • The Euclidean method employs a path integral formalism for calculating $\rho_V$. This involves structural techniques such as the replica trick, geometric interpretations, and conical singularity analysis. The entanglement entropy is computed using partition functions over replicated geometries, leading to analytical continuations that connect entropies for von Neumann and Rényi measures.
4. Real-Time Approach:
   • Utilizing Wick's theorem and canonical commutation/anticommutation relations, the real-time approach directly relates field correlators to the entanglement entropy. This method is instrumental in exploring interactions and non-local couplings in lattice field theories and discretized models.
5. Numerical Calculations and Analytical Examples:
   • The paper addresses both massive and massless field configurations across different dimensions, particularly through singular geometric shapes like polygons. Explicit calculations demonstrate universal terms, such as logarithmic contributions related to conical singularities, which are essential in even-dimensional spacetime theories.

Implications

The implications of this research are significant for both theoretical and applied physics. The understanding and computation of entanglement entropy play a crucial role in:

• Quantum Information Theory: The foundational results on entropy divergence provide insights into the information-theoretic properties of quantum systems, potentially impacting quantum computing and cryptography.
• Quantum Gravity and Black Holes: These computations aid in deciphering the holographic principle, aiding in the formulation of quantum gravity theories and understanding black hole thermodynamics.
• Condensed Matter Physics: The area law behavior and computational techniques are vital for analyzing quantum phase transitions and characterizing topological phases of matter.

Conclusion

By intertwining the Euclidean and real-time formalisms, this research provides a comprehensive framework for calculating entanglement entropy in free quantum field theories. It advances our understanding of quantum entanglement across various dimensions and physical contexts, paving the way for future research in both elementary particle physics and complex quantum system analyses. As theoretical tools improve and computational techniques evolve, these foundational insights will guide the paper of more intricate and interactive quantum systems.