An introduction to the theory of quantum entanglement

Quantum entangled states play very important roles in quantum information processing, such as quantum teleportation, dense coding, error correction, cryptography, and computation. The theory of entanglement mainly concerns the characterization of quantum entanglement and the classification of quantum states, as well as its applications in quantum information processing. We present some basic concepts and results, including: the relationship between locality and reality in quantum mechanics and Bell inequalities, and that between Bell inequalities and the separability of quantum states; the definition of separability for pure and mixed quantum states, and some separability criteria such as the positive map approach, positive partial transpose, reduction criterion, realignment, entanglement witnesses, and the covariance matrix approach and local uncertainty relations; some entanglement measures such as the entanglement of formation, concurrence, relative entropy, negativity, tangle, entanglement of assistance, and their estimation; the change in entanglement when the physical system evolves or undergoes interaction with the environment.