Probing Non-Markovianity through Topological Signatures of Path Space

The characterization of non-Markovian dynamics in open quantum systems remains a central challenge in quantum physics. Conventional measures often rely on specific properties of the system's evolution, such as the flow of information or the divisibility of the dynamical map. This paper introduces a fundamentally different approach by connecting quantum memory effects to the topological structure of the system's path space. We posit that the set of all possible quantum trajectories, when viewed as a geometric object, exhibits distinct topological signatures under non-Markovian evolution compared to its Markovian counterpart. Using tools from topological data analysis, specifically persistent homology, we analyze the path space generated from simulations of a canonical open quantum system. We demonstrate that topological invariants, such as the Betti numbers, which count holes and voids in the path space, serve as robust indicators of non-Markovianity. Our results reveal that the presence of quantum memory induces non-trivial topological features, such as persistent loops, in the space of paths. This suggests that memory effects constrain the system's evolution, forcing trajectories to organize into complex, non-contractible structures. This topological framework provides a model-agnostic and potentially more robust method for detecting and quantifying quantum memory, opening new avenues for understanding complex quantum dynamics from a geometric perspective.