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

TL;DR

This paper compares four different approaches to calculating the time-dependent complexity of thermofield double states, revealing both similarities and differences that deepen understanding of quantum complexity.

Contribution

It systematically analyzes and compares holographic and field theoretic complexity proposals, highlighting their behaviors at early and late times.

Findings

Early time complexity increases linearly in CV and FG proposals.

Late time growth rate saturates Lloyd's bound in CA, CV, and FG proposals.

FS proposal shows zero growth rate at late times.

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 $2E/(\pi\hbar)$ 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.