Complexity of Quantum Trajectories

TL;DR

Open quantum systems are governed by Lindblad dynamics, and ensembles of quantum trajectories offer a route to probe dynamical complexity beyond steady states. The authors introduce an unsupervised, data-driven measure—the intrinsic dimension $I_d$ of quantum trajectories—to quantify the effective dimensionality of trajectory manifolds in Hilbert space. Across driven-dissipative quantum tops and XXZ spin chains, $I_d$ exhibits minima at integrable points or in regimes with BBGKY decoupling/fragmentation, indicating reduced trajectory complexity even when the ensemble dynamics remains chaotic. The results bridge quantum chaos diagnostics and trajectory-based complexity, providing a robust tool that is largely insensitive to the particular unraveling and applicable to finite-size, strongly dissipative systems. This approach offers a new lens on ergodicity breaking and constrained open-system dynamics with potential implications for quantum simulation and control.

Abstract

Open quantum systems can be described by unraveling Lindblad master equations into ensembles of quantum trajectories. Here we investigate how the complexity of such trajectories is affected by conservation laws and other dynamical constraints of the underlying Lindblad evolution. We characterize this complexity using a data-driven approach based on the intrinsic dimension, defined as the minimal number of variables required to encode the information contained in a data set. Applying this framework to several systems, including dissipative variants of the quantum top and of the XXZ chain, we find that the intrinsic dimension is sensitive to the structure of their dynamics. The Lindblad evolution in these systems is typically chaotic, but additional constraints arise at specific parameter values, where the dynamics becomes integrable, exhibits Hilbert-space fragmentation, or develops a closed BBGKY hierarchy, leading to pronounced minima in the intrinsic dimension. Our approach results in an unsupervised probe of the complexity of dissipative quantum systems that is sensitive to chaos and ergodicity breaking phenomena beyond the initial transient regime.

Figures (15)

• Figure 1: Illustration of the scale dependence of the intrinsic dimension. If the nearest neighbours lie at a distance comparable to the scale of local noise, the structure appears two-dimensional. Only at an appropriately large scale the underlying curve emerges.
• Figure 2: Complex ratios of the GHS kicked top with parameters $S=80,\,\omega_z=4,\,g=5$, in the regular and chaotic regimes: $(a)$$\omega_x=0,\,k=0,\,\gamma=0.2$ (integrable); $(b)$$\omega_x=0\,, k=4,\,\gamma=4$ (with kicks); $(c)$ for $\omega_x=2,\,k=0,\, \gamma=2$. The level repulsion is well captured by the angular distributions, having (a) $\overline{\cos\theta}=0.001$, (b) $\overline{\cos\theta}=-0.23$, (c) $\overline{\cos\theta}=-0.24$.
• Figure 3: (a) Radial and (b) angular distributions corresponding to Fig. \ref{['fig: complex ratios']} (c), showing the convergence of the complex ratios to chaotic spectral statistics. The reference distributions are obtained by sampling $S=200$$N\times N$ random matrices with size $N=10000$ from the Ginibre ensemble.
• Figure 4: Chaotic behavior of the dissipative quantum top as seen from individual QT. We plot two QT of the model given by Eq. \ref{['eq: lindblad quantum top']} with $\omega_z=1, \,g=5,\,k=0$, considering in (a), (b) $\omega_x=0$, in (c), (d) $\omega_x=3$ and in (e) $\omega_x=5$. Even if trajectories are always perturbed by stochastic effects, the local structure between consecutive quantum jumps is one-dimensional in the integrable case (a). The subfigure (d) is a zoom on the spiraling region at the end of the trajectories in (a), showing that different trajectories are attracted towards the same one-dimensional region. Increasing $\omega_x$, a transient chaotic region starts to appear, where the trajectories get trapped before escaping to another long-time attractor. As shown in (e), this structure is not one-dimensional, in contrast to the integrable case. For sufficiently large values of $\omega_x$, the chaotic attractor takes over, as we depict in (e). The appearance of initially small chaotic regions that become dominant increasing the integrability-breaking coupling is reminiscent of what happens in the classical systems, where chaotic trajectories take over the phase space.
• Figure 5: Time evolution of the intrinsic dimension for the dissipative top with $\omega_z=1,\,g=5,\,\omega_x=4.0,\,k=0$ and $\gamma=2$, $S=30$. The value of $I_d(t)$ is estimated using different different numbers of trajectories, encoding now the scale dependence of $I_d$. With a larger $N$, the points are on average closer to each other and probe a finer scale of the data set, resulting in increasing values like for the $I_d$ calculated along individual trajectories.