The Leggett inequality is a constraint on the bipartite correlation that admits certain types of non-localities. Existing tests mainly focused on the electromagnetic systems where measurement apparatus are assumed to be projective and sharp. However, in nature there are interactions that do not obey the same conservation laws for photon, and the actual measurements may subject to unavoidable uncertainties due to the fundamental physical principles. In this work, we generalize the Leggett inequality to incorporate the measurements that are unsharp and/or biased. It is found that the parity violation in nature provides a spontaneous implementation of an unsharp measurement for the spin of hyperon. A fine structured Leggett inequality for hyperon decays characterized by the asymmetry parameters is obtained and its violation is found which could be observed with the yet obtained data in experiment, like BESIII and Belle.

22 pages, 7 figures; published in Phys. Rev. D