General Iterative Formulas of Bell's Inequalities.

Newton's second law aids us in predicting the location of a classical object after knowing its initial position and velocity together with the force it experiences at any time, which can be seen as a process of continuous iteration. When it comes to discrete problems, e.g. building Bell inequalities, as a vital tool to study the powerful nonlocal correlations in quantum information processing. Unless having known precisely the general formula of associated inequalities, iterative formulas build a bridge from simple examples to all elements in the set of Bell inequalities. Although exhaust all entities in the set, even in the subset of tight individuals, is a NP hard problem, it is possible to find out the evolution law of Bell inequalities from few-body, limited-setting and low-dimension situations to arbitrary $(n,k,d)$ constructions, i.e. $n$ particles, $k$ measurements per particle, and $d$ outcomes per measurement. In this work, via observing Śliwa's 46 tight (3,2,2) Bell inequalities [{Phys. Lett. A}. 317, 165-168 (2003)], uniting the "root" method [{Phys. Rev. A}. 79, 012115, (2009)] and the idea of degeneration, we discover an iterative formula of Bell inequalities containing all $(n,k,2)$ circumstances, which paves a potential way to study the current Bell inequalities in terms of iterative relations combining "root" method on the one hand, and explore more interesting inequalities on the other.