Complexity and Information in Quantum and Classical Trajectories

TL;DR

This work compares quantum jump trajectories of a driven, dissipative two-qubit system with a classical interacting telegraph-process analog to identify quantum signatures. Using $Lempel$-$Ziv$ complexity, mutual information, and temporal correlations, it shows that both models undergo a transition from independent to synchronized dynamics as the coupling $J$ grows, but only the quantum trajectories sustain enhanced complexity and substantial information sharing at large drive-to-decay ratios, with a strong $I$--$LZ$ coupling ($\rho \approx 0.82$). The classical model exhibits short-lived correlations and eventual freezing, whereas the quantum model maintains coherent fluctuations that keep trajectory structure nontrivial. Overall, trajectory-level statistics emerge as an effective diagnostic to distinguish quantum from classical dynamics in open systems and motivate extensions to larger networks and non-Markovian environments.

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.

Figures (4)

• Figure 1: Trajectory--level comparison of the classical interacting telegraph model and the quantum jump model for $\Omega=1.0$, and $\gamma=1.0$. (a) Representative emission trajectories for increasing coupling strength $J$. For weak coupling, both systems produce independent jump records. At intermediate $J$, intermittent bursts and correlated intervals appear, marking the onset of synchronized dynamics. For large $J$, the classical model locks into a frozen configuration, while the quantum model retains residual fluctuations due to coherence and measurement backaction. (b) Cumulative emission counts $N_{2}$ versus $N_{1}$. Weak coupling produces near--diagonal behavior consistent with independent activity. Increasing $J$ bends the curves into step--like structures, reflecting co--jumps and partial locking. Quantum trajectories remain smoother and less abrupt than the classical ones, showing the effect of quantum coherence in suppressing sharp switching. (c) Autocorrelation differences $\Delta C(\tau,J)=C(\tau,J)-C(\tau,0)$. Classical trajectories (left) develop slow, large--amplitude oscillations as $J$ increases, while quantum trajectories (right) remain more bounded and regular.
• Figure 2: Joint--state occupancy distributions for quantum and classical dynamics with the same parameters as in Fig. \ref{['fig:single_traj']}. Heatmaps show the joint probabilities $P(s_{1},s_{2})$ for $J=0,0.1,1$, and 3. The top row displays quantum and the bottom row shows classical results. At weak coupling, both models exhibit nearly uniform occupancy. Increasing $J$ enhances correlations and population asymmetry. For large $J$, classical trajectories collapse into a single dominant state, whereas the quantum system continues to explore all states through coherent fluctuations.
• Figure 3: Lempel--Ziv complexity of joint jump sequences in the uncoupled limit ($J=0$). (a) Complexity versus drive strength $\Omega$ at fixed decay rates $\gamma$. Increasing $\Omega$ enhances temporal structure, producing a complexity peak near $\Omega\sim\gamma$ before saturating at large drive. (b) Complexity versus decay rate $\gamma$ at fixed $\Omega$, showing the same crossover from coherent to incoherent emission approached from the opposite direction. (c) LZ complexity plotted against the ratio $\Omega/\gamma$ using logarithmic sampling. Both models collapse onto the same curve with a clear peak at $\Omega/\gamma\approx 1-2$, identifying the boundary between rate--limited and coherence--limited dynamics. Quantum curves lie slightly below classical ones at intermediate parameters due to measurement backaction. Peak LZ values: quantum 0.0257 $\pm$ 0.0018, classical 0.0248 $\pm$ 0.0016; $t =$ 0.375, $p =$ 0.71.
• Figure 4: (a) Lempel?Ziv (LZ) complexity versus coupling strength $J$ for quantum (dashed) and classical (dotted) models for different drive--to--decay ratios $\Omega/\gamma$. Increasing coupling suppresses randomness and drives both systems toward structured, synchronized dynamics, with the quantum model retaining higher complexity at large $\Omega/\gamma$. (b) Mutual information versus $J$ for the same parameters. Quantum mutual information increases with $\Omega/\gamma$ and peaks at intermediate coupling, reflecting coherence--assisted information exchange, while classical mutual information peaks at smaller $\Omega/\gamma$ and decreases at strong coupling due to stochastic synchronization and freezing. (c) Mutual information plotted against LZ complexity, with points colored by coupling strength $J$. The quantum model exhibits a strong monotonic correlation between information and complexity, whereas the classical model shows a weaker, non--monotonic relation, indicating fundamentally different mechanisms for correlation generation. Spearman coefficient: quantum $\rho = 0.817$, $p = 4.0 \times 10^{-25}$; classical $\rho = 0.162$, $p = 0.11$.