Mixed State Entanglement Entropy in CFT (2501.08198v1)

Abstract: How to calculate the entanglement entropy between two subsystems for a mixed state has remained an important problem. In this paper, we provide a straightforward method, namely the subtraction approach, to solve this problem for generic covariant bipartite mixed states, where time dependence is explicitly included. We further demonstrate that the mixed state entanglement entropy $S_\text{vN}$ can be calculated in a more abstract yet powerful way. Within the context of the AdS$3$/CFT$_2$ and AdS$_5$/CFT$_4$ correspondences, we show that $S\text{vN}$ exactly matches the corresponding entanglement wedge cross section in the AdS bulk, respectively.

• The paper introduces the subtraction approach to calculate mixed state entanglement entropy in CFTs, showing the von Neumann entropy aligns with the entanglement wedge cross-section in AdS.
• The subtraction method is extended to dynamic, generic configurations and higher dimensions (CFT4) using conformal transformations and renormalization group flows.
• Results reinforce the holographic correspondence between calculated entropies and the covariant entanglement wedge cross-section, offering insights into quantum entanglement and spacetime structure.

Overview of Mixed State Entanglement Entropy in Conformal Field Theory

The paper "Mixed State Entanglement Entropy in CFT" by Xin Jiang, Peng Wang, Houwen Wu, and Haitang Yang addresses the intricate problem of quantifying entanglement entropy in mixed states within conformal field theories (CFTs). This work is particularly focused on the development of the "subtraction approach" to calculate entanglement entropy between two subsystems, encapsulating time dependence and providing insights within the framework of AdS/CFT correspondence.

Key Contributions and Methodology

The traditional measures of entanglement entropy, effective for pure states, prove inadequate for mixed states such as two disjoint intervals in CFT$_{2}$. The authors offer the subtraction method, an alternative to purification techniques, circumventing the complexity of introducing auxiliary systems. This approach involves subtracting undetectable regions of spacetime, thereby reducing the system to a configuration that represents a pure entangled state.

For symmetric configurations, they leverage the replica trick combined with annulus CFT techniques to calculate the von Neumann entropy, $S_\text{vN}$, for two disjoint intervals at $t=0$. The study notably finds that $S_\text{vN}(A:B)$ aligns perfectly with the entanglement wedge cross-section (EWCS), $E_W(A:B)$, within a static time slice of the AdS$_3$ space.

Extension to Covariant and Higher-Dimensional CFTs

The analysis extends to covariant scenarios by incorporating time dynamics, thus rendering the subtraction method applicable to generic configurations of intervals defined by endpoints with non-zero time parameters. By invoking conformal transformations, they map complex, asymmetric configurations onto symmetric setups on a transformed plane, facilitating the computation of von Neumann entropy based on determined annular regions.

For higher-dimensional theories, specifically CFT$_{4}$, the researchers utilize renormalization group (RG) flows to define the entanglement entropy across renormalization scales. The derived RG flow-induced entanglement entropy is shown to coincide with the EWCS in the context of the AdS$_5$/CFT$_4$ duality, bolstering the universality and applicability of their method across dimensions.

Holographic Interpretations and Implications

The paper further elaborates on the holographic dual of mixed state entanglement entropy, aligning $S_\text{vN}(A:B)$ with the covariant entanglement wedge cross-section in AdS spaces. This connection crucially accounts for the gauge parameters inherent in determining bulk-boundary point correlations, reinforcing the correspondence between holographic computations and boundary conformal data.

Implications for Future Research

The results firm up the foundational understanding of quantum entanglement in mixed states, elucidating the intricate geometrical relations prescribed by the AdS/CFT framework. The techniques lay a groundwork for further exploration into non-trivial entanglement structures within higher-dimensional and dynamic spacetime configurations. The application of this framework could significantly advance the understanding of quantum gravity phenomena, offering a refined lens through which the emergent structure of spacetime might be scrutinized.

Conclusion

Overall, this work enriches the computational toolkit available for entanglement quantification in mixed states, augmenting both theoretical paradigms and holographic inferences in quantum field theories. The alignment of calculated entropies with entanglement wedge constructs not only affirms theoretical predictions but also propels the refinement of techniques employed in the holistic calculation of quantum entanglement across varied dimensionalities and symmetric configurations.