Abstract
Programmable quantum simulators are beginning to access correlators of increasing complexity, ranging from four-point out-of-time-ordered correlators to even higher-order many-body correlators. The theoretical framework for interpreting such data, however, remains comparatively underdeveloped. Although a variety of higher-order correlators can be constructed straightforwardly by changing the ordering, timing, or choice of operators, their physical meaning is often difficult to infer. A further complication is that different correlators are generally not independent: some may be mutually redundant, while others may encode genuinely distinct many-body information. These features make it necessary to analyze correlators not as isolated quantities, but as a structured family. In this work, we develop a geometric framework for the collective analysis of higher-order correlator families. By representing correlators as inner products between operator words, we recast each family as a geometry in operator space. The key idea is to introduce conditioning subspaces that separate this geometry into reducible information, already explained by a chosen resolved sector, and irreducible information, encoded in the residual correlator geometry. Focusing on the latter component, we define irreducible volume profiles that quantify how broadly the unexplained part of a correlator family spreads over independent geometric directions. This perspective leads to several complementary forms of conditioning. Canonical conditioning optimally explains a correlator family, revealing that free-fermion, interacting integrable, and chaotic spin-chain dynamics generate qualitatively different correlator geometries. Targeted conditioning instead fixes the resolved sector to isolate a chosen physical feature, allowing us to identify the spatial confinement of correlator geometry under many-body localization, characterize the organization of measurement-inaccessible correlator components, and reveal state-dependent structure in correlator geometry from a spectral perspective. Finally, Krylov and cross conditioning extend the framework from a single correlator family to comparisons among correlator geometries: the former tracks how the geometry evolves in time, while the latter identifies irreducible structures that are shared or reshaped across different dynamics. Our framework reveals irreducible structures hidden at the level of individual correlator values and establishes correlator geometry as a higher-level description of quantum many-body dynamics.
