Contextuality from the projector overlap matrix

Official URL: https://dx.doi.org/10.1088/1751-8121/ae8f8c

We develop a geometric formulation of quantum contextuality organized around the overlap matrix (Formula presented) (Formula presented) in a (Formula presented) (Formula presented) -dimensional Hilbert space, where (Formula presented) (Formula presented) and (Formula presented) (Formula presented) are the joint-eigenspace projectors of two pairs of compatible observables sharing a common observable. The matrix (Formula presented) (Formula presented) quantifies the information shared between these joint-eigenspaces. Rather than characterizing contextuality through outcome statistics alone, this formulation analyzes the geometric incompatibility encoded in (Formula presented) (Formula presented). Two distinct scalar contractions of (Formula presented) (Formula presented) summarize this geometric content: the mutual information energy (Formula presented) (Formula presented), and the Maassen–Uffink parameter (Formula presented) (Formula presented), the largest amplitude overlap between vectors from the two joint eigenbases. We work with the logarithmic form (Formula presented) (Formula presented) (the subscript indicates Rényi order (Formula presented) (Formula presented) ) throughout for its additivity and dimensional compatibility with standard entropic measures. Applied to Klyachko–Can–Binicioğlu–Shumovsky (KCBS) and Clauser—Horne—Shimony—Holt (CHSH) scenarios, the framework identifies regimes in which every state-dependent witness considered here is silent yet (Formula presented) (Formula presented) by an amount set by the projector geometry alone. We prove that (Formula presented) (Formula presented) is non-increasing under coarse-graining of the projector families, that (Formula presented) (Formula presented) is a necessary configuration-level condition for observable contextuality, and that for a configuration (Formula presented) (Formula presented) decomposed into context pairs (Formula presented) (Formula presented), the additive composition (Formula presented) (Formula presented) is exact in both the KCBS and CHSH scenarios. The resulting picture provides a state-independent geometric signature of contextuality that complements outcome-based and Shannon-entropic witnesses.