On the Time Dependence of Holographic Complexity

Abstract: We evaluate the full time dependence of holographic complexity in various eternal black hole backgrounds using both the complexity=action (CA) and the complexity=volume (CV) conjectures. We conclude using the CV conjecture that the rate of change of complexity is a monotonically increasing function of time, which saturates from below to a positive constant in the late time limit. Using the CA conjecture for uncharged black holes, the holographic complexity remains constant for an initial period, then briefly decreases but quickly begins to increase. As observed previously, at late times, the rate of growth of the complexity approaches a constant, which may be associated with Lloyd's bound on the rate of computation. However, we find that this late time limit is approached from above, thus violating the bound. Adding a charge to the eternal black holes washes out the early time behaviour, i.e., complexity immediately begins increasing with sufficient charge, but the late time behaviour is essentially the same as in the neutral case. We also evaluate the complexity of formation for charged black holes and find that it is divergent for extremal black holes, implying that the states at finite chemical potential and zero temperature are infinitely more complex than their finite temperature counterparts.

• The paper finds that holographic complexity shows distinct temporal patterns under the CA and CV conjectures in eternal black hole settings.
• It employs numerical analysis to demonstrate a constant growth rate in uncharged black holes and a smoother transition in charged cases.
• The study highlights potential violations of Lloyd’s Bound and calls for reevaluating complexity models in quantum gravity research.

Analyzing the Dynamics of Holographic Complexity in Black Holes

The paper "On the Time Dependence of Holographic Complexity" presents a detailed investigation into the dynamics of holographic complexity within various eternal black hole backgrounds, employing both the complexity=action (CA) and complexity=volume (CV) conjectures. As a foundational aspect of the study, the authors explore the implications of these frameworks within the paradigms of the AdS/CFT correspondence, focusing on the thermofield double state as a dual description of two-sided black holes.

A key area of investigation is the temporal evolution of complexity in the context of uncharged versus charged black holes, where the CA and CV conjectures predict notably distinct behaviors. The study reveals that, within the CV conjecture, the complexity monotonically increases with time, asymptotically approaching a constant value. Contrarily, in the CA scenario, the complexity initially remains static before a rapid increase, a pattern attributed to specific characteristics of the Wheeler-DeWitt patch's interaction with the past singularity.

The authors provide an important highlight on the numerical results, notably that the complexity, when analyzed through the CV conjecture, surges at a constant growth rate dependent on holographic quantities like mass and charge distribution in black holes. The analysis concludes that this rate correlates with the Lloyd’s computational bound.

Detailed calculations address the impact of these ideas on charged black holes, emphasizing that the presence of charge modifies the initial holographic complexity behavior. Notably, the addition of charge smooths the transition from initial inaction to rapid complexity growth. This observation in the CA framework introduces potential violations to Lloyd’s Bound, as complexity growth surpasses expected thresholds, inviting further discourse on the universality and constraints of this boundary in quantum computations.

The investigation extends to a rigorous exploration of charged case complexities, illuminating an intriguing divergence in complexity formation for extremal black holes. The divergence suggests an infinite escalation in complexity compared to finite temperature states—a proposal resonant with the third law of thermodynamics analogies within complexity theory.

The comprehensive analysis provided unveils several theoretical nuances, such as the dependence of late-time configurations on the null normalization constants, drawing attention to questions about the holographic description's robustness and universal applicability. The authors evoke a call for revisiting assumptions inherent in complexity models, with implications for both theoretical advancements and more grounded computational analogs.

Looking forward, this work lays the groundwork for future studies to explore the holographic complexity by extending beyond black hole paradigms to potentially encompass different spacetime configurations or coupling regimes, such as those involving rotatory dynamics or non-commutative properties. The exploration into computational and physical equivalences presents itself as fertile ground for potential breakthroughs in understanding quantum gravity's computational facets.

In conclusion, "On the Time Dependence of Holographic Complexity" provides a substantial addition to our understanding of holographic complexity, intersecting foundational quantum information concepts with intricate spacetime geometry. It challenges existing models while raising foundational questions about our conceptual grasp of complexity in quantum systems, setting a pivotal reference for ongoing research in theoretical physics.