Comparison of holographic and field theoretic complexities by time dependent thermofield double states

TL;DR

The paper analyzes time-dependent complexity for thermofield double states using four proposals: two holographic (complexity=volume CV and complexity=action CA) and two field-theoretic (Fubini-Study FS and Finsler geometry FG). It introduces a renormalized holographic complexity framework via complexity potentials and computes the evolution of complexity under CA and CV, finding late-time linear growth with a rate set by the total energy, and highlighting parameter choices that satisfy the Lloyd bound. In parallel, it constructs time-dependent TFD states in free field theory and evaluates FS and FG complexities, showing FS generally yields decreasing/vanishing late-time growth while FG produces early-time linear growth and can be tuned to saturate Lloyd’s bound but with subleading violations. Overall, CV and FG display closer qualitative alignment, suggesting a deeper connection between holographic CV and field-theoretic FG, and the study clarifies how different definitions of complexity influence time evolution and bounds.

Abstract

We compute the time-dependent complexity of the thermofield double states by four different proposals: two holographic proposals based on the "complexity-action" (CA) conjecture and "complexity-volume" (CV) conjecture, and two quantum field theoretic proposals based on the Fubini-Study metric (FS) and Finsler geometry (FG). We find that four different proposals yield both similarities and differences, which will be useful to deepen our understanding on the complexity and sharpen its definition. In particular, at early time the complexity linearly increase in the CV and FG proposals, linearly decreases in the FS proposal, and does not change in the CA proposal. In the late time limit, the CA, CV and FG proposals all show that the growth rate is $2E/(π\hbar)$ saturating the Lloyd's bound, while the FS proposal shows the growth rate is zero. It seems that the holographic CV conjecture and the field theoretic FG method are more correlated.

Figures (8)

• Figure 1: Penrose diagrams and WDW patches for AdS$_{d+1}$ black hole ($d\geq3$) when $|t_R|<\Delta t_c$ (left panel) and $|t_R|>\Delta t_c$ (right panel). In the left panel, the past null sheets will meet the singularity at $B_1$ and $B_2$ respectively. In the right panel, the past null sheets will meet each other at $B$ with $r=r_0\in(0,r_h)$.
• Figure 2: The complexity $\mathcal{C}(|\text{TFD}(0,t_R)\rangle,|0\rangle)$ and its growth rate when $d>2$. $\mathcal{C}_0$ is the complexity when $t_R=\Delta t_c$ and $\dot{\mathcal{C}}_m=2M/\pi\hbar$. Higher dimensional cases give the similar results.
• Figure 3: The complexity $\mathcal{C}(|\text{TFD}(0,t_R)\rangle,|0\rangle)$ and their growth rates for the BTZ black hole. $\mathcal{C}_0$ is the complexity when $t_R=0$ and $\dot{\mathcal{C}}_m=2M/\pi\hbar$.
• Figure 4: Extremal surfaces in the AdS black hole. For given time slices at the left and right boundary, the volume of extremal surface connecting these two time slices (blue curve) gives the holographic complexity potential. The upper red dotted line is for $\tilde{t}_B=\infty$ and the middle red dotted line is for $\tilde{t}_B=0$.
• Figure 5: The values of $\mathcal{C}(|\text{TFD}(T,t_L+t_R)\rangle,|0\rangle)$ and its growth rate when $d=2,3,4,5$. The higher dimensions give similar results. $\mathcal{C}_0$ is the complexity when $t_L+t_R=0$ and $\dot{\mathcal{C}}_f$ is the Lioyd's bound of growth rate, which is given by Eq. \ref{['dCdtAdS2CV']}.