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  • Open Access

Complexity of Quantum Trajectories

Luca Lumia1,2, Emanuele Tirrito3,4, Mario Collura1,5, Fabian H. L. Essler6, and Rosario Fazio7,8

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PRX Quantum 7, 033049 – Published 9 September, 2026

DOI: https://doi.org/10.1103/pbjx-fwqn

Abstract

Open quantum systems can be described by unraveling their Lindblad master equation 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 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.

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Popular Summary

When a quantum system interacts with its environment through noise, loss, or continuous measurement, its evolution can be described in two complementary ways. One is the density matrix, which averages over all possible outcomes. The other is an ensemble of quantum trajectories: individual, stochastic “stories” of how the system’s state evolves under a particular sequence of environmental interactions. The density matrix obeys a Lindblad master equation, whose dynamics can display features associated with quantum chaos or integrable behavior. In classical non-linear dynamics, integrable systems have regular one-dimensional orbits, while chaotic trajectories are attracted toward complicated subsets called “strange attractors” with fractal geometry and non-integer dimension. Inspired by this view, we ask whether the dynamical complexity of Lindblad evolution is reflected in the structure of the trajectories themselves. We introduce a data-driven notion of complexity based on the intrinsic dimension: the effective number of independent variables needed to represent the information contained in a trajectory data set. Intuitively, it asks how many “coordinates” are required to faithfully describe what the trajectories are doing, linking naturally theoretic limits on information compressibility and to geometric methods used to characterize attractor-like sets. Applying this approach to paradigmatic quantum models, we find that the intrinsic dimension reliably tracks dynamical structure, showing minimal complexity when additional constraints such as integrability simplify the master equation. Crucially, this signal persists beyond transients: after the density matrix relaxes to a steady state and becomes largely insensitive to dynamical details, trajectories can still populate a structured subset of state space whose effective dimension can be inferred directly from data.

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PRX Quantum 7, 033049– Published 9 September, 2026
  • Received 18 February 2026
  • Revised 21 May 2026
  • Accepted 29 June 2026
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