Complexity from the Reduced Density Matrix: a new Diagnostic for Chaos

TL;DR

The paper tackles diagnosing quantum chaos in multiparticle/open quantum systems by developing a circuit-complexity based diagnostic. It centers on a reduced-density-matrix construction via the Jamiołkowski operator-state mapping to form an effective target state, and compares this approach to full-system complexity and complexity of purification using a two-oscillator toy model with inverted dynamics. The main finding is that complexity derived from the reduced density matrix captures scrambling time $t_s$ and Lyapunov growth dominated by the bath parameter (and can reveal the full Lyapunov spectrum), whereas the complexity of purification is less sensitive. This work offers a natural open-system chaos diagnostic and motivates extensions to more realistic chaotic models such as spin chains, highlighting the role of environment in chaotic dynamics.

Abstract

We investigate circuit complexity to characterize chaos in multiparticle quantum systems. In the process, we take a stride to analyze open quantum systems by using complexity. We propose a new diagnostic of quantum chaos from complexity based on the reduced density matrix by exploring different types of quantum circuits. Through explicit calculations on a toy model of two coupled harmonic oscillators, where one or both of the oscillators are inverted, we demonstrate that the evolution of complexity is a possible diagnostic of chaos.

Figures (7)

• Figure 1: Left: Complexity for single evolved (in cyan) and doubly evolved (in blue) with $\epsilon_1=-5, \epsilon_2=-5, \lambda=0.1, \delta \lambda =0.001$ and Right: Complexity for single evolved (in cyan) and doubly evolved (in blue) with $\epsilon_1=-1, \epsilon_2=1, \lambda=0.1, \delta \lambda =0.001.$
• Figure 2: Left: Complexity for forward-backward evolved target state for perturbation in different directions with $\epsilon_1= -6, \epsilon_2=-1, \lambda=0.1$ Cyan: $\delta\epsilon_1=0.001, \delta\epsilon_2=0, \delta\lambda=0$, Blue: $\delta\epsilon_1=0, \delta\epsilon_2=0.001, \delta\lambda=0$, Black: $\delta\epsilon_1=0, \delta\epsilon_2=0, \delta\lambda=0.001$, Red: $\delta\epsilon_1=0.001, \delta\epsilon_2=0.001, \delta\lambda=0$. Right: Complexity display different Lyapunov when $\epsilon$ parameters for the system and bath oscillators are different. Here we took $\epsilon_1=-6, \epsilon_2=-2, \lambda=0.1, \delta \epsilon=0.001, \delta \lambda=0.001$ The two slopes are displayed by red and blue dashed lines.
• Figure 3: Complexity display two separate linear growths when the parameters ($\epsilon_1, \epsilon_2$) for the system and bath oscillators are different. In the left panel the bath parameter $\epsilon_2$ is fixed at -6 and the system parameter $\epsilon_1$ is scanning from -1 to -6. Whereas in the right panel, the system parameter $\epsilon_1$ is chosen as -6, -7, -10, -12 and -20 from dashed cyan to solid cyan curve respectively.
• Figure 4: Complexity for the effective wave function when both oscillators are inverted (left panel) and when one of the oscillators are inverted. The parameters are -- Left:$\epsilon_1= - 5.01, \epsilon_2= - 5, \lambda=0.1$ and Right: black dashed curve has $\epsilon_1= 5.01, \epsilon_2=- 5, \lambda=0.1$ and the blue curve has $\epsilon_1= -5.01, \epsilon_2= 5, \lambda=0.1$.
• Figure 5: Complexity for $\epsilon_2=-5, \lambda=0.1$