Comments on Holographic Complexity

Abstract: We study two recent conjectures for holographic complexity: the complexity=action conjecture and the complexity=volume conjecture. In particular, we examine the structure of the UV divergences appearing in these quantities, and show that the coefficients can be written as local integrals of geometric quantities in the boundary. We also consider extending these conjectures to evaluate the complexity of the mixed state produced by reducing the pure global state to a specific subregion of the boundary time slice. The UV divergences in this subregion complexity have a similar geometric structure, but there are also new divergences associated with the geometry of the surface enclosing the boundary region of interest. We discuss possible implications arising from the geometric nature of these UV divergences.

• The paper examines both the CV and CA conjectures, detailing their geometric interpretations and UV divergence structures in holographic complexity.
• It highlights that the CV conjecture faces arbitrary scaling challenges while the CA conjecture overcomes these with a refined treatment of null boundary terms.
• The analysis extends to subregion complexity, addressing normalization concerns and linking holographic complexity to practical quantum information applications.

Comments on Holographic Complexity: A Review

The paper under review explores the notion of holographic complexity, a topic gaining considerable attention at the intersection of quantum information science, quantum field theory, and quantum gravity. Specifically, the authors focus on two conjectures in the context of the AdS/CFT correspondence: the "complexity=volume" (CV) conjecture and the "complexity=action" (CA) conjecture. These conjectures propose geometric dualities in AdS/CFT, connecting the complexity of quantum states, as understood in quantum information theory, to certain geometrical quantities in the AdS bulk.

Overview and Results

1. CV Conjecture: According to the complexity=volume conjecture, the complexity of the boundary state is dual to the volume of an extremal codimension-one hypersurface in the AdS bulk. While this conjecture provides a straightforward geometrical interpretation, it involves an arbitrary scale choice, such as the AdS curvature scale or the black hole horizon radius, which is dissatisfactory from a theoretical standpoint. The authors analyze the ultraviolet (UV) divergences in these volume calculations and show that the leading divergences scale with the volume of the boundary CFT, while subleading divergences involve integrals of curvature invariants.
2. CA Conjecture: The complexity=action conjecture, on the other hand, equates complexity with the action evaluated on the Wheeler-DeWitt (WDW) patch in the bulk. Unlike the CV conjecture, the CA conjecture does not have arbitrary parameters, but initially lacked a rigorous method to handle null boundary terms in the gravitational action. This was recently resolved, allowing for detailed analysis. The paper investigates the structure of UV divergences for CA duality, noting a similar dependence on geometric quantities as in CV duality but with additional terms related to logarithmic divergences.

Implications and Theoretical Insights

• Geometric Nature of Complexity: One of the most striking findings is the geometric nature of the intricacies of complexity, manifest in the UV divergences of holographic complexity. These divergences, the authors argue, are indicative of the role of complexity in encoding local correlations in the boundary CFT down to small scales.
• Subregion Complexity: The paper extends the analysis of holographic complexity to subregions, proposing that the associated complexity should be computed using the CA or CV rules over the entanglement wedge of the subregion. This extends the holographic dictionary to mixed states and brings holographic complexity closer to practical quantum information applications.
• Normalization and Interpretation Challenges: Throughout the work, the authors raise concerns regarding the normalization factors involved in CA duality and the physical interpretations of these terms. The subtleties of defining complexity extend beyond the mathematical formalism to conceptually understanding what these holographic prescriptions mean for quantum information.

Future Directions

The paper identifies several avenues for further exploration, including:

• Understanding the implications of these divergences for quantum error correction properties of holographic states.
• Investigating the impact of higher curvature corrections in the gravitational dual, which may refine the relationships between geometric complexity and field-theoretic definitions.
• Exploring connections to other holographic constructs, such as entanglement entropy, and investigating potential new holographic duals for different types of quantum complexity metrics.

This exploration provides a fertile ground for theoretical development and presents intriguing questions about the nature of space-time, quantum information, and the fundamental underpinnings of the holographic principle. The work presented forms a key component of the growing literature on holographic complexity, offering both methodological refinements and deeper theoretical insights into this complex and multi-faceted domain.