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arXiv:2512.19848

Complexity and Information in Quantum and Classical Trajectories

Hira Ali, Naeem Shahid

quant-phSubmitted Dec 22, 2025• Announced Dec 24, 2025• 0 citations
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Abstract

We analyze emission trajectories from a driven-dissipative two-qubit system and a classical telegraph model with matched rates. Using Lempel-Ziv complexity, mutual information, and temporal correlations, we show that both models undergo a transition from independent to synchronized dynamics as coupling increases, but only the quantum trajectories develop enhanced complexity and sustained information sharing at large drive-to-decay ratio. Classical correlations are short-lived and quickly suppressed by strong drive. A strong complexity-information correlation appears uniquely in the quantum case, providing a clear trajectory-level signature of quantum effects. These results show that complexity and information measures extracted directly from jump records provide an efficient way to distinguish quantum and classical dynamics in open systems.

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We analyze emission trajectories from a driven-dissipative two-qubit system and a classical telegraph model with matched rates. Using Lempel-Ziv complexity, mutual information, and temporal correlations, we show that both models undergo a transition from independent to synchronized dynamics as coupling increases, but only the quantum trajectories develop enhanced complexity and sustained information sharing at large drive-to-decay ratio. Classical correlations are short-lived and quickly suppressed by strong drive. A strong complexity-information correlation appears uniquely in the quantum case, providing a clear trajectory-level signature of quantum effects. These results show that complexity and information measures extracted directly from jump records provide an efficient way to distinguish quantum and classical dynamics in open systems.
We investigate signatures of non-Markovianity in the dynamics of a periodically driven qubit coupled to a dissipative bosonic environment. We propagate the dynamics of the reduced density matrix of the qubit by integrating the numerically exact hierarchical equations of motion. Non-Markovian features are quantified by comparing on an equal footing the predictions from diverse and complementary approaches to quantum dissipation. In particular, we analyze the distinguishability of quantum states, the decay of the volume accessible to the qubit on the Hilbert space, the negativity of the canonical rates in the generalized Lindblad equation, and the relaxation of the memory kernels in the Nakajima-Zwanzig generalized quantum master equation. We study the effects of controlled driving on the coherent dynamics of the system. We show that a suitable external field can offset the ergodic relaxation of time-correlation functions, increase distinguishability over time, and strengthen non-Markovian effects witnessed by the canonical dissipation channels. We furthermore observe the phenomenon of eternal non-Markovianity for sufficiently small system-bath coupling and we discuss how this can be enhanced by modulating the frequency of the external drive. This work provides a broad theoretical analysis of quantum dissipation in the framework of open quantum dynamics and quantum information.
The notion of complexity pervades physics, with manifestations rang- ing from classical networks to quantum many-body dynamics. This thesis investigates the complexity of open quantum systems, focus- ing on the interplay between unitary evolution and dissipation, with particular emphasis on measurement processes. Two complementary perspectives are developed: an approach based on quantum resource theories and a data-driven analysis of quantum trajectories. In the first part, quantum complexity is characterized in terms of resources such as entanglement, non-stabilizerness, and non-Gaussianity. These resources capture the non-classical features that enable quantum com- putational advantage. We analyze the generation of resources in mon- itored systems, where the competition between coherent dynamics and measurements gives rise to novel phase transitions. Through semiclassical analyses, tensor networks, and Gaussian and stabilizer techniques, we identify distinct dynamical regimes and clarify the interplay between different resources under repeated quantum mea- surements. The second part focuses on a classical complexity measure inspired by non-parametric machine learning methods, known as the intrinsic dimension. We employ this measure to quantify the com- plexity of ensembles of quantum trajectories, demonstrating that it is sensitive to structural properties such as integrability, fragmenta- tion, and the decoupling of correlation functions, thereby witnessing the complexity of the underlying dynamics. By combining resource- theoretic and unsupervised learning approaches, this thesis provides a wide perspective on the complexity of monitored quantum systems, bridging statistical physics, quantum information, and data science.
The ubiquitous effects of the environment on quantum-mechanical systems generally cause temporally correlated fluctuations. This particularly holds for systems of interest for quantum computation where such effects lead to correlated errors. The Markovian approximation neglects these correlations and thus fails to accurately describe open-system dynamics where these correlations become relevant. In driven open systems, yet another approximation is persistently used, often unknowingly, in which one describes the decay effects independently from the time-dependent controlling fields acting on the system, thereby ignoring further temporally correlated effects. To overcome these shortcomings, we develop a quantum master equation for driven systems weakly coupled to quantum environments that avoids the aforementioned field-independent approximation, as well as the Markovian approximation. Our method makes it possible to track all occurring decay channels and their time-dependent generalized rates which we illustrate in the example of a generally driven two-level system. We also demonstrate that correlated and field-dependent dissipative effects can lead to an increase in the performance of single-qubit gate operations.
We employ a quantum trajectory approach to characterize synchronization and phase-locking between open quantum systems in nonequilibrium steady states. We exemplify our proposal for the paradigmatic case of two quantum Van der Pol oscillators interacting through dissipative coupling. \nWe show the deep impact of synchronization on the statistics of phase-locking indicators and other correlation measures defined for single trajectories, spotting a link between the presence of synchronization and the emergence of large tails in the probability distribution for the entanglement along trajectories. Our results shed new light on fundamental issues regarding quantum synchronization providing new methods for its precise quantification.
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We consider a parametrically driven Kerr medium in which the pumping may be sinusoidally varied. It has been previously found that this system exhibits coherent tunneling between two fixed points which can be either enhanced or suppressed by altering the driving frequency and strength. We numerically investigate the dynamics when damping is included. This is done both by solving a master equation and using the quantum-trajectory method. In the latter case it is also possible to model the result of a continuous heterodyne measurement of the cavity output. The dissipation destroys the coherences which give rise to the tunneling, causing the sinusoidal oscillation of the mean to give way to a stochastic jumping between the fixed points, manifested as a random telegraph signal. In the quantum-trajectory picture we show that the coherences responsible for tunneling are an exponentially decreasing function of the signal-to-noise ratio for heterodyne measurements. However, evidence of both the bare tunneling rate and the driving modified tunneling rate are still apparent in the random telegraph signal.