Quantum nonlocality of anomalous weak values

Violations of Bell inequalities serve as a signature of entanglement, with only entangled states capable of exceeding classical bounds. Here, we analyze a standard Bell test characterized by violations of the Clauser–Horne–Shimony–Holt (CHSH) inequality in post-selected quantum settings, focusing on the maximal weak values that can arise under various constraints. We interpret these values as generalized bounds that extend the notion of nonlocal correlations to post-selected systems. We consider three distinct scenarios: (1) both the pre- and post-selected states are unrestricted, leading to a Tsirelson-type bound for post-selected correlations; (2) one of the states is constrained to be separable; and (3) both states are restricted to be separable. To ensure a fair comparison, we fix the magnitude of the overlap between the pre- and post-selected states across all cases. For each scenario, we derive two types of bounds. First, we determine the maximal weak value for a fixed Bell operator, revealing how the separability of the states impacts the achievable correlations with post-selection. Second, we compute the maximal bounds by jointly optimizing over both the states and the Bell operator, thereby identifying the highest possible correlations attainable for a given overlap in each case. Our results demonstrate that entanglement remains essential for exceeding generalized classical bounds, even with post-selection. Moreover, we find that entanglement and post-selection can act as a joint resource, enabling stronger-than-classical correlations beyond what either can produce alone. These findings clarify the distinct roles of post-selection and entanglement in nonlocality and suggest new possibilities for quantum sensing and computation using post-selection.