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

TLDR

The study computes the time‑dependent complexity of thermofield double states using four approaches: holographic CA and CV conjectures and field‑theoretic Fubini‑Study and Finsler geometry methods. The four proposals show both commonalities and divergences: early‑time complexity rises linearly in CV and FG, falls linearly in FS, and remains flat in CA, while at late times CA, CV and FG saturate Lloyd’s bound (2E/πħ) whereas FS stays constant, indicating that the holographic CV and field‑theoretic FG methods are most closely aligned.

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/(πℏ) 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.