On the Time Dependence of Holographic Complexity

TL;DR

This paper studies the time evolution of holographic complexity in black hole backgrounds using CA and CV conjectures, revealing monotonic growth, late-time saturation, and violations of Lloyd's bound, with effects of charge and extremality.

Contribution

It provides a detailed analysis of the full time dependence of holographic complexity, including late-time behavior, charge effects, and complexity of formation for extremal black holes.

Findings

CV complexity rate increases monotonically and saturates.

CA complexity remains constant initially, then decreases briefly, then increases, violating Lloyd's bound.

Extremal black holes have divergent complexity of formation.

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.