Entanglement detection via the condition of quantum correlation

We develop a necessary condition of quantum correlation. It is utilized to construct a d-level bipartite Bell-type inequality which is strongly resistant to noise and requires only analyses of O(d) measurement outcomes compared to the previous result O(d2). Remarkably, a connection between the arbitrary high-dimensional bipartite Bell-type inequality and entanglement witnesses is found. Through the necessary condition of quantum correlation, we propose that the witness operators to detect truly multipartite entanglement for a generalized Greenberger-Horne-Zeilinger (GHZ) state with two local measurement settings and a four-qubit singlet state with three settings. Moreover, we also propose a robust entanglement witness to detect a four-level tripartite GHZ state with only two local measurement settings