Understanding of complex quantum systems advances by revealing how excess complexity, a measure of dynamical heterogeneity, is determined in composite structures. For systems built from tensor products of interacting parts, this leading complexity arises from a weighted variance of Lanczos coefficients calculated for each individual part; key to this is that the same measurement also defines contributions to both mismatch and entropy within the system. Complexity in interconnected quantum systems clarifies as variations in calculations performed on individual components dictate overall system complexity, while also defining randomness within the structure.

Excess complexity, a measure of dynamic differences, links to fluctuations in Lanczos coefficients calculated for each part of a composite system. This finding establishes new ways to probe collective behaviour and understand intricate interactions between quantum parts. Calculations performed on individual components dictate overall system complexity and define randomness within the structure; this builds upon understanding of ‘excess complexity’, which measures dynamic differences between parts.

Excess complexity, a measure of dynamical heterogeneity, links to fluctuations in ‘Lanczos coefficients’ calculated for each part of a composite system. Consider Lanczos Coefficients as weighting factors in an average grade, where some assignments matter more than others when calculating the final result. This finding establishes new ways to probe collective behaviour, particularly in structures built from independent “Lego brick” subsystems combined via a tensor product where they do not interfere with one another. But increasing the number of these interconnected components affects overall complexity and reveals emergent behaviours within the quantum structure?

Lanczos Coefficient Variance Quantifies Complexity Beyond Synchronous Evolution

Previous measures of dynamical heterogeneity required identifying when composite quantum systems evolved *in synchrony*. Now, scientists at University of Cape Town demonstrate that leading excess complexity, the measurement of dynamic differences, is controlled by a weighted variance of Lanczos coefficients calculated for each subsystem.

This represents an advancement over previous methods which struggled with asynchronous subsystems by shifting focus from observing interactions to quantifying internal fluctuations. Determining if parts were evolving together previously demanded detailed knowledge about their combined Hamiltonian and Krylov decomposition; this new approach establishes the same coefficient variance governs both mismatch contributions to complexity *and* entropy, offering complementary ways to probe collective behaviour without direct observation of inter-component interaction.

Further analysis revealed accurate prediction of excess complexity even when individual subsystem Hamiltonians are non-identical, a scenario where earlier methods faltered due to asynchronous behaviour. The scientists demonstrated consistent correlations between internal fluctuations and overall system heterogeneity across multiple tensor product configurations. While pinpointing quantifiable relationships is valuable, translating them into predictive power for real-world complex systems remains challenging because modelling truly large numbers of interacting degrees of freedom presents limitations; it’s an open question whether this approach will scale effectively as systems become more intricate.

Quantifying temporal variation in isolated quantum dynamics

University of Cape Town scientists demonstrate a powerful new set of tools for quantifying complexity within neatly compartmentalised quantum systems, those built as tensor products where subsystems don’t interfere. Their findings deliberately sidestep entanglement, raising the question of how well this measure holds up when components *do* interact strongly. Acknowledging that this work excludes strong linking between quantum components doesn’t diminish its value, however, establishing a baseline understanding for more complex systems and quantifying internal change over time.

Fluctuations in calculations performed on each part reveal that the complexity arising from combining multiple quantum systems is fundamentally linked to internal variations within individual components. This shifts understanding away from needing detailed knowledge of interactions, a key simplification for analysing large structures, towards quantifying inherent differences instead. Establishing excess complexity and Krylov entropy as complementary tools offers researchers two distinct yet related ways to investigate collective behaviour without directly observing interactions, paving the way for future research extending this measure to include entangled components and explore truly active dynamics.

Researchers demonstrated a relationship between the complexity of combined quantum systems, built from tensor products of subsystems, and fluctuations within those individual parts. The study showed that leading complexity is controlled by variance in subsystem calculations, offering a new quantitative method for measuring dynamical heterogeneity and synchronous evolution. This work establishes both excess complexity and Krylov entropy as useful methods for investigating how collective behaviours emerge in large quantum systems; authors suggest further investigation could extend these measures to more complex, interacting systems.

👉 More information
🗞 Krylov Complexity and Entropy of Composite Quantum Systems
✍️ Jeff Murugan, Victor Van Aswegen and Hendrik J R Van Zyl (University of Cape Town)
🧠 ArXiv: https://arxiv.org/abs/2610.31394

More like this