Complementary Local and Nonlocal Measurements of Conjugate Transformation - Unified Logic Representation of Conjugate 0-1 Vectors

Abstract

Bohr proposed the complementarity principle in 1927 as the foundation of quantum mechanics, since then relevant debates have been critically discussed for many years. Applying a pair of spin particles, Einstein proposed the EPR paradox in 1935. Using nonlocal potential properties, Aharonov and Bohm proposed the AB effect to test complementary measurement results. Under locality conditions, Bell established a Bell inequality under classical logic. Using a pair of particles and double sets of ZMI devices for complementarity measurements, Hardy proposed the Hardy paradox in 1992. During the past 50 years, locality and nonlocality tests on complementarity were hot-topics among the advanced quantum information, computing and measurement directions with various theoretical extensions and solid experimental results.These complementarity approaches separated local/nonlocal parameters to form different equations without an integrated logic framework to describe these equations including both local and nonlocal features consistently. The main results provide a series of paradoxes that conflict with each other. This paper uses conjugate transformation. Based on the m+1 kernel form of 0-1 states, n pairs of conjugate partitions were established. Under a given configuration in N bits, a set of 2n 0-1 feature vectors are applied to construct conjugate transformation operators in logic levels with intrinsic measurements to be a set of measurement operators.The key results of the paper are listed in Theorem 5. Two special functions of vector logic (CNF or DNF expression) and four equivalent expressions of the elementary equation are examples to show local and nonlocal variables in equations consistently. Applying two pairs of conjugate sets <A, B> and their complementary sets <A', B''>, 4 meta measures are established corresponding to <±aA;±bB> quantitative features under measurement operators. The main results of the paper are represented in Lemma (1-4), Theorem (1-5) and Corollary (1-7). From a vector logic viewpoint, conjugate complementary scheme can organize local and nonlocal variables to satisfy the comprehensive properties of modern logic constructions on completeness, non-conflict and consistence in a united logic framework.