Does Complexity Equal Anything?

Abstract: We present a new infinite class of gravitational observables in asymptotically Anti-de Sitter space living on codimension-one slices of the geometry, the most famous of which is the volume of the maximal slice. We show that these observables display universal features for the thermofield-double state: they grow linearly in time at late times and reproduce the switch-back effect in shock wave geometries. We argue that any member of this class of observables is an equally viable candidate as the extremal volume for a gravitational dual of complexity.

• The paper introduces a new infinite class of gravitational observables that generalize the complexity-volume and complexity-action proposals.
• The paper reveals universal features including linear growth over time and the switchback effect in shock wave geometries.
• The paper employs diffeomorphism-invariant methods to establish robust connections between gravitational observables and quantum complexity.

An Overview of Gravitational Observables in Holographic Complexity

The paper "Does Complexity Equal Anything?" by Belin et al. delineates the exploration of gravitational observables within asymptotically Anti-de Sitter (AdS) space, focusing on their relevance to the concept of complexity in quantum systems. The authors propose a new infinite class of observables related to gravitational systems that extend beyond existing models such as the complexity-volume (CV) and complexity-action (CA) proposals.

Universal Features and Observables in AdS

This study extends the discussion surrounding quantum complexity, particularly in the context of black holes and holography. In this framework, the complex structure of gravitational duals is examined through observables defined on codimension-one slices of the geometry. The paper emphasizes that these observables display universal features, notably a linear growth over time and the manifestation of the switchback effect in shock wave geometries. These features have previously been associated with metric volume proposals, but the authors introduce a broader class of viable candidates that can serve as gravitational duals for holographic complexity.

Definition and Motivation

The authors define observables $O_{F_1, \Sigma_{F_2}}$ based on scalar functions $F_1$ evaluated over a surface $\Sigma_{F_2}$. This approach generalizes prior methods by allowing diverse functionals within a diffeomorphism-invariant framework. The diffeomorphism invariance is a crucial property, ensuring these observables genuinely reflect quantum complexity rather than relying on a particular metric representation.

Analytical Approach and Results

The paper methodically analyzes the properties of these observables under specific states, such as the thermo-field double state, which is vitally significant in entanglement entropy discussions in quantum gravity. The linear growth of the observables corroborated the predicted behavior of complexity in black hole thermodynamics and further supported the switchback effect. This effect is particularly indicative of the complexity growth disruptions caused by adding operator insertions that pertain to shock waves in holographic duals.

Significance and Potential Implications

This research broadens the horizon of holographic complexity by challenging the existing paradigms limited to the volume or action on specific geometrical slices. By presenting a plethora of observables that could logically embody complexity, the study highlights the innate ambiguities and choices inherent in defining complexity, akin to gate set choices in quantum computational theory.

Future Prospects

The prospects for future development are extensive, encompassing further investigation into codimension-0 observables and their interaction with boundary conditions in the dual CFT. Additionally, producing a more comprehensive correlation between the ambiguities in quantum complexity definitions and the broad class of gravitational observables remains a crucial area of inquiry. The work opens avenues for embedding these observables into coherent frameworks to elucidate bulk-boundary correlations in gravitational theories.

In conclusion, Belin et al.'s paper sets a significant milestone in the field of theoretical physics by expanding the registry of gravitational observables pertinent to holographic complexity. Despite provocations and ambiguities in these models, the potential theoretical and practical implications attest to the rich interplay between holography and complexity in quantum gravity.