Causality and Duality in Multipartite Generalized Probabilistic Theories

Causality is one of the most fundamental notions in physics. Generalized probabilistic theories (GPTs) and the process matrix framework incorporate it in different forms. However, a direct connection between these frameworks remains unexplored. By demonstrating the duality between no-signaling principle and classical processes in tripartite classical systems, and extending some results to multipartite systems, we first establish a strong link between these two frameworks, which are two sides of the same coin. This provides an axiomatic approach to describe the measurement space within both box world and local theories. Furthermore, we describe a logically consistent 4-partite classical process acting as an extension of the quantum switch. By incorporating more than two control states, it allows both parallel and serial application of operations. We also provide a device-independent certification of its quantum variant in the form of an inequality.