Controlled NOT function can provoke biased interpretation from Bell’s test experiments

Abstract

Abstract. Recently, we showed that the controlled-N OT function is a permutation that cannot be inverted in subexponential time in the worst case [Quantum Information Processing. 16:149 (2017)]. Here, we show that such a condition can provoke biased interpretations from Bell's test experiments.Let CN OT be the canonical two-qubit entangling gate in quantum key distribution (QKD) cryptographic protocols, where CN OT |a, x = |a, a + x , so that the control parameter a and the target variable x ∈ F 2 = {0, 1}.For x = a, CN OT |a, x = |a, x 2 + x , since x ∧ x = x = x 2 , and for x = a, CN OT |a, x = |a,(i) The permutation x 2 +x = x⊕x is a factorable polynomial (reducible) over a finite field of two elements, whose Hamming distance between its even inputs is equal to 0 (local model), and (ii) The permutation x 2 + x + 1 = x ⊕ N OT (x) is a nonfactorable polynomial (irreducible) over a finite field of two elements, whose Hamming distance between its odd inputs is not equal to 0 (nonlocal model). However, these models are deducible from each other because x 2 + x (+1) = 0 (+1) = x 2 + x + 1 andConsider the Hadamard basis {|+ , |− } of a one-qubit register given by:The circuit below takes computational basis F 2 = {0, 1} to Bell states: