Because of the importance and the fundamental role of entanglement in quantum information pro- cessing and quantum computing, a lot of researches have been done on entanglement generation, variation and degradation in various domains. . Recently, relativistic quantum information process- has attracted a lot of interests . The world is fundamentally relativistic, therefore, understanding entanglement in space-time is ultimately important. It is realized that relativity plays a significant role in quantum entanglement and related quantum protocols, such as quantum teleportation. We apply the Schwinger pair production theory to constant and pulsed electric fields on a Minkowski spacetime to demonstrate that the background electric field can generate the entanglement. We work out the entanglement entropy for scalar particles and Dirac fermions created by the background electric field. Also, we show that the positive scalar curvature of the de Sitter spacetime in the absence of an electric field can generate entanglement while the negative scalar curvature of the anti-de Sitter degrades the generated entanglement. We show that the expanding universe can generate the entanglement entropy. Investigation of the generated entanglement in an expanding universe for bosonic and fermionic par- ticles shows some deep differences between these modes. Finally, we study quantum teleportation with nonclassical correlated state in noninertial frame