Complexity and Shock Wave Geometries

Abstract: In this paper we refine a conjecture relating the time-dependent size of an Einstein-Rosen bridge to the computational complexity of the of the dual quantum state. Our refinement states that the complexity is proportional to the spatial volume of the ERB. More precisely, up to an ambiguous numerical coefficient, we propose that the complexity is the regularized volume of the largest codimension one surface crossing the bridge, divided by $G_N l_{AdS}$. We test this conjecture against a wide variety of spherically symmetric shock wave geometries in different dimensions. We find detailed agreement.

• The paper establishes that quantum state complexity is proportional to the regularized volume of an ER bridge via a refined complexity-volume conjecture.
• It employs spherically symmetric shock wave geometries to rigorously test the conjecture and reveal chaotic dynamics through switchback mechanisms.
• The study implies that long-time quantum complexity encodes black hole interior structures and advances holographic models in quantum gravity.

Analyzing the Relationship Between Complexity and Einstein-Rosen Bridge Geometry

The paper by Douglas Stanford and Leonard Susskind examines the intricate link between the computational complexity of quantum states and the geometry of Einstein-Rosen bridges (ERBs) within the AdS/CFT framework. The authors refine a prior conjecture, proposing an association between the spatial volume of an ERB and the complexity of its dual quantum state. Specifically, the paper conjectures that the complexity is proportional to the regularized volume of the maximal codimension-one surface spanning the ERB, divided by fundamental constants $G_N$ and $l_{AdS}$, with broad compatibility observed across various shock wave geometries.

Background and Conjecture

The ERB of an eternal AdS black hole is a feature that classically grows indefinitely, contrasting with the quick equilibration expected in dual boundary theories. Despite thermal equilibrium halting classical evolution, quantum complexity continues to evolve, serving as a time-extended descriptor of states. Previously, complexity was linked to the ERB's length, but this study posits spatial volume as a more precise correlate, formalized in conjecture (1). The paper suggests evaluating ERB volume through codimension-one surface maximization, contrasting from Ryu-Takayanagi-style length approaches limited by existence concerns in asymmetrical configurations.

Analytical Framework and Results

The authors rigorously test the complexity-volume conjecture through spherically symmetric shock wave geometries, constructing configurations by temporal operator insertions into an eternal AdS black hole. They find detailed agreement with complexity scaling predicted by the conjecture. The complexity of a perturbed state shows a specific dependence on the folding structure of time in its preparation, capturing chaotic dynamics through partial cancelation mechanisms at switchback points.

Implications and Future Directions

The paper reveals profound implications for understanding black hole interiors and holographic complexity. By using volume as a proxy for complexity, it suggests that long-time quantum properties encode information about bulk geometries past equilibrium points, important for encoding physics past event horizons. The study hints at potential further exploration in more general geometries, including non-spherical perturbations, and considers constraints upon complexity through circuit depths and tensor network models.

Conclusion

Stanford and Susskind's work provides evidence supporting a nuanced understanding of holographic complexity's relation to spatial volume in ERBs, turning a conjecture into a foundation for interpreting quantum chaotic processes in holography. It emphasizes complexity's role in comprehending black hole microstates and their time-dependent features, advancing potential theoretical developments in quantum gravity's ongoing discussions.