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In most Bell tests, the measurement settings are specially chosen so that themaximal quantum violations of the Bell inequalities can be detected, or atleast, the violations are strong enough to be observed. Such choices canusually associate the corresponding Bell operators to a kind of effectiveobservables, called pseudo Pauli operators, providing us a more intuitiveunderstanding of Bell nonlocality in some sense. Based on that, a more generalquantum-to-classical approach for the constructions of Bell inequalities isdeveloped. Using this approach, one can not only derive several kinds ofwell-known Bell inequalities, but also explore many new ones. Besides, we showthat some quadratic Bell inequalities can be induced from the uncertaintyrelations of pseudo Pauli operators as well, which may shed new light on thestudy of uncertainty relations of some nonlocal observables.

Bell Inequalities Induced by Pseudo Pauli Operators on Single Logical Qubits