Position dependence of Nielsen complexity for the Thermofield double state

TL;DR

This study explores how the complexity of a thermofield double state in a harmonic oscillator depends on position, frequency, and external electric fields using Nielsen's geometric approach, revealing intricate behaviors under varying conditions.

Contribution

It introduces an analysis of position-dependent Nielsen complexity for thermofield double states, highlighting the effects of external electric fields and frequency variations with numerical insights.

Findings

Complexity decreases with increasing frequency.

External electric fields alter the complexity behavior.

Higher electric fields restore previous complexity patterns.

Abstract

In this paper, the Nielsen geometric method is used to study the position dependence of the Nielsen complexity for the thermofield double state of a harmonic oscillator. We present the state shift under the influence of an external electric field and demonstrate its importance for the construction of the corresponding circuit. By numerical analysis, we investigate the effect of the frequency and the external field on the dynamics of complexity. Our observation reveals that the system's complexity diminishes considerably with the rise of the frequency. Furthermore, our findings indicate that the complexity exhibits a distinct behavior under a feeble external electric field, as it grows more intricate with the escalation of the frequency. However, with higher magnitudes of the electric field, the system reverts to its prior behavior. We also remark on the influence of the reference state's frequency on the complexity.