Quantum Chaos, Complexity and Decoherence

What will the state of quantum technologies be in thirty years? Will we be able to overcome decoherence in complex devices that exhibit quantum chaotic behaviour? Or, similar to the limitations in a Carnot heat engine, should we accept that the optimal extraction of quantum resources is fundamentally constrained? Exploring quantum chaos in open many-body quantum systems presents significant challenges. A key issue is that the time evolution of dissipative quantum systems is typically governed by non-Hermitian dynamical generators. As a result, their eigenvalues cannot be systematically ordered, hindering the use of spectral statistics. Additionally, studying many-body quantum chaos is challenging because the spectra of generic interacting systems are essentially intractable. While quantum simulators could offer valuable insights, they are always prone to some level of environmental noise and provide limited access to spectral information. In contrast, dynamical measures such as correlation functions and fidelities can be feasibly traced in experimental settings. This thesis conducts an extensive exploration of theoretical tools to tackle quantum complexity in the presence of decoherence. We focus on generalizing the Spectral Form Factor (SFF) as the survival probability (SP) of an initial Coherent Gibbs State (CGS), examining its behavior and properties in various scenarios of open dynamics. This approach is suitable for both open and many-body systems. We study a series of dissipative models, including Lindbladian dynamics, non-Hermitian systems, and quantum channels. We further introduce a framework to directly relate time-dependent manifestations of quantum chaos and coherence monotones, linking quantum chaos to the resource theory of coherence. Coherence is perceived as a necessary ingredient of quantum chaos while quantum noise is always suppressed by decoherence. In complex quantum systems, it is fruitful to search for common patterns present in systems that are randomly sampled but share some symmetry. Our focus in most numerical examples is on random matrix calculations to test the universality of our approach.
This requires studying the average behavior over different systems or in finite intervals of time evolution. A key discovery is the linkage of time and ensemble averages of the SFF to unitarity breaking. The thesis addresses the long-standing question regarding the self-averaging nature of the SFF, demonstrating its self-averaging property at long timescales in open quantum systems. Furthermore, it probes the properties of the newly introduced notion of Krylov complexity, revealing its connection to the SFF. These insights collectively advance the understanding of quantum chaos in many-body systems, opening pathways for further exploration of quantum complexity in the presence of decoherence.