On entanglement spreading in chaotic systems (1608.05101v3)

Abstract: We discuss the time dependence of subsystem entropies in interacting quantum systems. As a model for the time dependence, we suggest that the entropy is as large as possible given two constraints: one follows from the existence of an emergent light cone, and the other is a conjecture associated to the "entanglement velocity" $v_E$. We compare this model to new holographic and spin chain computations, and to an operator growth picture. Finally, we introduce a second way of computing the emergent light cone speed in holographic theories that provides a boundary dynamics explanation for a special case of entanglement wedge subregion duality in AdS/CFT.

An Analysis of Entanglement Spreading in Chaotic Systems

Mark Mezei and Douglas Stanford's paper titled "On entanglement spreading in chaotic systems" explores the intricate phenomena related to the time evolution of subsystem entropies within interacting quantum systems. The paper introduces a theoretical framework addressing the constraints on entanglement spread, particularly focusing on the entanglement velocity $v_E$ and the emergent light cone speed $v_{LC}$.

Key Aspects

1. Definitions and Parameters: The paper discusses three speeds crucial to understanding entanglement spreading:
   • Entanglement Velocity ($v_E$): Defined as the early-time rate of entropy growth per area after a quench.
   • Light Cone Speed ($v_{LC}$): Represents the effective speed at which information commutes between spatially separated operators.
   • Butterfly Velocity ($v_B$): The speed at which information represented by operator commutators spreads throughout the system.

The authors suggest that $v_{LC}$ and $v_B$ are often identical, although exceptions may occur, indicating differing slopes in information spread dynamics.

1. Constraints on Entropy Growth: The paper proposes two main constraints:
   • The entanglement growth rate is constrained by the region's area multiplied by the entanglement velocity.
   • Information must adhere to an emergent light cone defined by $v_{LC}$.

These bounds interplay in a complex manner to model the subsystem entropy's time dependence, and the paper compares these theoretical models to data from holographic systems and chaotic spin chains.

1. Numerical Comparisons and Holography: The analysis finds that holographic systems can saturate the entropy as rapidly as the constraints allow, aligning closely with these theoretical bounds. This saturation hints at a model where chaotic systems with shared $v_{LC}$ and $v_B$ capture coarse-grained features of information dynamics via a combined bound.
2. Future Implications: The results have implications for understanding chaotic spin chains, and provide an operator growth model as a potential explanation for the entanglement dynamics observed. The paper also relates the butterfly speed to concepts in holographic theories, such as the entanglement wedge in the AdS/CFT correspondence, thus suggesting deep connections between chaotic systems and holographic principles.

Observations and Implications

• Practical and Theoretical Insights: This paper provides insights into practical applications related to quantum information spread and theoretical implications of subsystem dynamics. The approximations to modeling constraints offer a better understanding of entropy evolution and shed light on the speed at which information traverses through chaotic systems.
• Challenges for Future Research: The authors highlight the need to refine the distinction between $v_{LC}$ and $v_B$, particularly in systems where these speeds differ, such as certain spin chains and Bose-Hubbard models. Understanding these discrepancies could further unravel the nuances of quantum information dynamics.
• Connections to Holography: The association of the butterfly velocity with holographic entanglement wedge concepts suggests that chaotic systems might adopt similar reconstructive dynamics as observed in holographic models, paving the way for unified theories that bridge quantum chaos and gravity.

Overall, Mezei and Stanford's work emphasizes the complexity and richness of entanglement spreading dynamics in chaotic systems and the relevance of considering diverse speeds and constraints in theoretical models. By comparing these dynamics with data from both spin models and holographic computations, they offer a compelling narrative on how chaotic systems may approach and potentially saturate hypothesized bounds.