Holographic local quench and effective complexity

TL;DR

This work analyzes how holographic complexity, computed via both the CV and CA conjectures, evolves after a local quench in AdS3/CFT2, comparing it with entanglement entropy and entanglement density. Using a point-like falling particle as the holographic quench, the authors derive analytical and perturbative results for total-system and subsystem complexities, revealing distinct early-time growth, nontrivial subsystem behavior, and Lloyd bound saturation under CA. The findings suggest that holographic complexity encodes aspects of effective, non-random structure formation, with CV and CA highlighting complementary features and offering a bridge to notions of physical complexity. The study points to future directions in more complex inhomogeneous quenches and potential links to network-based representations of quantum states.

Abstract

We study the evolution of holographic complexity of pure and mixed states in $1+1$-dimensional conformal field theory following a local quench using both the "complexity equals volume" (CV) and the "complexity equals action" (CA) conjectures. We compare the complexity evolution to the evolution of entanglement entropy and entanglement density, discuss the Lloyd computational bound and demonstrate its saturation in certain regimes. We argue that the conjectured holographic complexities exhibit some non-trivial features indicating that they capture important properties of what is expected to be effective (or physical) complexity.

Figures (11)

• Figure 1: The density of the renormalized volume form $\Sigma$ for $M=0.2$ and $\alpha=0.5$. The left plot corresponds to the time moment $t=3$, and the right one to $t=10$.
• Figure 2: On the left plot, we show the time dependence of the (rescaled) complexity $\Delta {\cal C}$ of the excitation. On the right plot, we present the dependence $\Delta {\cal C}(t)$ on (rescaled) $\Delta S(t)$ as time changes from $t=0$ to $t=10$. The green curves correspond to $\alpha=0.25$, the blue ones -- to $\alpha=0.5$, and the red ones -- to $\alpha=1$.
• Figure 3: On the left plot we show the time dependence of the (rescaled) complexity of the excitation $\Delta {\cal C}$ for interval $x\in(-\ell,\ell)$ and for $\alpha=0.5$. Different curves correspond to different $\ell=2,3,4,5$ for down to top. On the right plot the same for fixed $\ell=2$ and $\alpha=0.25,0.5,0.75,1$ from down to top.
• Figure 4:
• Figure 5: Left plot: time dependence of the integrated entanglement density $\mathcal{N}$ (solid) and the volume complexity (dashed) of the interval $x\in (-3,3)$ at $M=0.2,\,\alpha=0.5$. Right plot: the relation between the (rescaled) subsystem volume complexity $\Delta C$ and $\mathcal{N}$.