Entanglement criterion and strengthened Bell inequalities based on the Pearson correlation

Entanglement is a property associated with quantum correlations and represents a key resource in several applications of quantum technology. Therefore, the ability to characterize entanglement is important at both foundational and practical levels. This work demonstrates how the Pearson correlation coefficient can be used to establish an entanglement criterion for quantum systems of two qubits. This criterion is then used to prove that a proposed conjecture is correct for the case of two qubits, which allows to efficiently identify entanglement without the need of complete prior knowledge of the quantum state. For higher dimensional quantum states the conjecture is demonstrated to be false through counter-examples, therefore a modified version of it is proposed. Finally, two new strengthened Bell inequalities are derived, which are also efficient in entanglement identification.