Self‐Guided Methods on Complex Variables for Searching the Maximal Violation of an Instrumental Test

ABSTRACT Quantum tests directly reveal the incompatibility of quantum and classical predictions. The violation of such a test establishes a quantifiable gap between the capabilities of observers with access to classical resources and those utilizing quantum entanglement. Typically, larger violations indicate a higher degree of quantumness—for instance, enabling quantum self‐testing in Bell scenarios, where achieving maximal violation is highly desirable. The self‐guided method, initially developed for finding maximal Bell violations, is later extended to complex variables. Here, we apply and compare both variants to determine the maximal violation of the instrumental inequality. The complex‐number variant generally exhibits faster convergence, except during the initial phase. This improvement is more evident in instrumental tests, likely due to their simpler causal structure compared to Bell scenarios. Our findings establish the self‐guided method with complex variables as a generally applicable and competitive approach for robustly and efficiently identifying maximal violations across diverse quantum tests.