The quantum nonlocal correlation of quantum states plays an important role in the quantum information and quantum computing protocols. However, during the transmission of entangled states in the quantum channel, they will inevitably interact with the environment, resulting in the degradation of the coherence and then weakening the quantum nonlocal correlation. Using a high probability quantum nonlocal correlation testing scheme based on Hardy-type paradox, in this paper we investigate the quantum nonlocal correlation testing of two-qubit polarization entangled states when they transmit through amplitude damping channel (ADC), phase damping channel (PDC) and depolarization damping channel (DC). The results show e to overcome the successful quantum nonlocal correlation testing of ADC by using weak measurement and quantum weak measurement reversal. The results show that when the intensity of weak measurement increases, the influence of ADC decoherence effect on quantum nonlocal correlation testing can be effectively reduced.

Quantum nonlocal correlation is one of the important features that distinguish the quantum theory from classical theory. As a typical quantum mixed state, the study of quantum nonlocal correlation based on the "X" state is of great importance for the verification of the correctness of quantum theory and the application of quantum information theory. In this paper, combined with the traditional Clauser-Horne-Shimony-Holt (CHSH) inequality testing for quantum nonlocal correlations, we propose a strategy for testing the quantum nonlocal correlations based on the geometric interpretation of the "X" state. By using the geometric interpretation of the "X" state, which is described by the transform of Bloch sphere, it is possible to investigate the optimal selection of measurement settings. The maximum value of CHSH inequalities also can be obtained from the physical images. Finally, the range of parameters for a successful quantum nonlocal correlation testing based on the CHSH inequality for the "X" state is studied. The results show that, when <i>f</i>=1, the "X" state will reduce to the normal pure entangled state, and the quantum nonlocal correlation testing results are in full agreement with the traditional ones. This result proves the correctness of the geometric interpretation strategy proposed in this paper. When<i>f</i><1, only some of the "X" states can be used for the successfully testing of quantum nonlocal correlations. It is also found that, the range of fidelity values that can be successfully testing the quantum nonlocal correlations will be further increased by increasing the values of <i>r</i>. In particular, when <i>r</i>=1, the range of fidelity value will reach the largest one (e.g. <i>f</i>>0.781). The results in this paper can provide the reference for the experimental testing of the quantum nonlocal correlation by using the "X" state.

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<<i>F</i><1, the maximum value <i>S</i>_<i>m</i> of testing quantum nonlocality will gradually transition from <i>S</i>_<i>m</i>>2 to <i>S</i>_<i>m</i><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 <i>F</i>>1. In particular, when <i>F</i>=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 ${\rho _W}$.