A Complexity for Quantum Field Theory States and Application in Thermofield Double States

TL;DR

This paper introduces a new complexity measure for quantum field theory states using a Finsler structure based on ladder operators, and applies it to thermofield double states, revealing finite complexity proportional to temperature and linking it to fidelity susceptibility.

Contribution

It defines a novel complexity concept for quantum field theory states using Finsler geometry and applies it to thermofield double states, connecting complexity with fidelity susceptibility and holographic duality.

Findings

Complexity density is finite and proportional to T^{d-1}.

Fidelity susceptibility equals the complexity between TFD and vacuum states.

Provides insights into holographic conjectures of complexity.

Abstract

This paper defines a complexity between states in quantum field theory by introducing a Finsler structure based on ladder operators (the generalization of creation and annihilation operators). Two simple models are shown as examples to clarify the differences between complexity and other conceptions such as complexity of formation and entanglement entropy. When it is applied into thermofield double (TFD) states in $d$-dimensional conformal field theory, results show that the complexity density between them and corresponding vacuum states are finite and proportional to $T^{d-1}$, where $T$ is the temperature of TFD state. Especially, a proof is given to show that fidelity susceptibility of a TFD state is equivalent to the complexity between it and corresponding vacuum state, which gives an explanation why they may share the same object in holographic duality. Some enlightenments to holographic conjectures of complexity are also discussed.