Quantum nonlocality testing of the “X” state based on the CHSH inequality in Markov Environment

Quantum nonlocality is one of the most fundamental characteristics of quantum theory. As a commonly used quantum state generated in experiments, the "X" state is a typical one in the research of open quantum systems, since it still maintains the stability of the "X" shape during the evolution. Using the Clauser-Horne-Harmony-Holt (CHSH) inequality, the quantum nonlocality testing of two "X" states associated with local transformation operations is studied under the Markov environment. The results show that in the phase damping environment, the two "X" states have the same CHSH inequality testing results with the increasing of the evolution time. Moreover, the maximum of quantum nonlocality test of the two "X" states will decrease nonlinearly. When 0.78<F<1, the maximum value Sm of testing quantum nonlocality will gradually transition from Sm>2 to Sm<2 with the increasing of the evolution time of the two "X" states, and the research on the quantum nonlocality test cannot be successfully carried out. In the amplitude damping environment, using the "X" state obtained by the local transformation operation have a longer evolution time for the successfully quantum nonlocality testing when F>1. In particular, when F=1, the "X" state with the density matrix $\rho _W$ cannot successfully perform the quantum nonlocality testing after the evolution time $\Gamma t > 0.22$. For the "X" state with density matrix ${\tilde \rho _W}$, the quantum nonlocality testing cannot be performed until the evolution time $\Gamma t > 0.26$. This results show that the local transformation operation of the "X" state is more conducive to the quantum nonlocality testing based on the CHSH inequality. Finally, the fidelity ranges of successfully testing the quantum nonlocality of the two "X" states in phase and amplitude damping environments are given in detail. The results show that, on the premise of quantum nonlocality testing successfully, the two types of "X" states evolving in the phase damping environment have the large range of valid fidelity. Meanwhile, at the same evolution time, the local transformation operation is helpful to improve the fidelity range of quantum nonlocality test in amplitude damping environment for "X" state with density matrix ${\rho _W}$.