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 experiment, the “X” state is a typical one in the research of open quantum systems, since it remains 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 increase of the evolution time. Moreover, the maximum of quantum nonlocality test of the two “X” states will decrease nonlinearly. When &lt;inline-formula&gt;&lt;tex-math id="M1"&gt;\begin{document}$0.78 \lt F \lt 1$\end{document}&lt;/tex-math&gt;&lt;alternatives&gt;&lt;graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="5-20222218_M1.jpg"/&gt;&lt;graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="5-20222218_M1.png"/&gt;&lt;/alternatives&gt;&lt;/inline-formula&gt;, the maximum value &lt;inline-formula&gt;&lt;tex-math id="M2"&gt;\begin{document}${S_m}$\end{document}&lt;/tex-math&gt;&lt;alternatives&gt;&lt;graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="5-20222218_M2.jpg"/&gt;&lt;graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="5-20222218_M2.png"/&gt;&lt;/alternatives&gt;&lt;/inline-formula&gt; of testing quantum nonlocality will gradually transition from &lt;inline-formula&gt;&lt;tex-math id="M3"&gt;\begin{document}${S_m} \gt 2$\end{document}&lt;/tex-math&gt;&lt;alternatives&gt;&lt;graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="5-20222218_M3.jpg"/&gt;&lt;graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="5-20222218_M3.png"/&gt;&lt;/alternatives&gt;&lt;/inline-formula&gt; to &lt;inline-formula&gt;&lt;tex-math id="M4"&gt;\begin{document}${S_m} \lt 2$\end{document}&lt;/tex-math&gt;&lt;alternatives&gt;&lt;graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="5-20222218_M4.jpg"/&gt;&lt;graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="5-20222218_M4.png"/&gt;&lt;/alternatives&gt;&lt;/inline-formula&gt; with the increase 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, the “X” state obtained by the local transformation operation has a longer evolution time for successfully testing quantum nonlocality when &lt;inline-formula&gt;&lt;tex-math id="M5"&gt;\begin{document}$F \gt 0.78$\end{document}&lt;/tex-math&gt;&lt;alternatives&gt;&lt;graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="5-20222218_M5.jpg"/&gt;&lt;graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="5-20222218_M5.png"/&gt;&lt;/alternatives&gt;&lt;/inline-formula&gt;. In particular, when &lt;inline-formula&gt;&lt;tex-math id="M6"&gt;\begin{document}$F = 1$\end{document}&lt;/tex-math&gt;&lt;alternatives&gt;&lt;graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="5-20222218_M6.jpg"/&gt;&lt;graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="5-20222218_M6.png"/&gt;&lt;/alternatives&gt;&lt;/inline-formula&gt;, the “X” state with the density matrix &lt;inline-formula&gt;&lt;tex-math id="M7"&gt;\begin{document}${\rho _W}$\end{document}&lt;/tex-math&gt;&lt;alternatives&gt;&lt;graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="5-20222218_M7.jpg"/&gt;&lt;graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="5-20222218_M7.png"/&gt;&lt;/alternatives&gt;&lt;/inline-formula&gt; cannot successfully test the quantum nonlocality after the evolution time &lt;inline-formula&gt;&lt;tex-math id="M8"&gt;\begin{document}$\varGamma t \gt 0.22$\end{document}&lt;/tex-math&gt;&lt;alternatives&gt;&lt;graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="5-20222218_M8.jpg"/&gt;&lt;graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="5-20222218_M8.png"/&gt;&lt;/alternatives&gt;&lt;/inline-formula&gt;. For the “X” state with density matrix &lt;inline-formula&gt;&lt;tex-math id="M9"&gt;\begin{document}${\tilde \rho _W}$\end{document}&lt;/tex-math&gt;&lt;alternatives&gt;&lt;graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="5-20222218_M9.jpg"/&gt;&lt;graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="5-20222218_M9.png"/&gt;&lt;/alternatives&gt;&lt;/inline-formula&gt;, the quantum nonlocality testing cannot be performed until the evolution time &lt;inline-formula&gt;&lt;tex-math id="M10"&gt;\begin{document}$\varGamma t \gt 0.26$\end{document}&lt;/tex-math&gt;&lt;alternatives&gt;&lt;graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="5-20222218_M10.jpg"/&gt;&lt;graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="5-20222218_M10.png"/&gt;&lt;/alternatives&gt;&lt;/inline-formula&gt;. These 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 the successful testing of quantum nonlocality , the two types of “X” states evolving in the phase damping environment have a large range of valid fidelity. Meanwhile, for the same evolution time, the local transformation operation is helpful in improving the fidelity range of quantum nonlocality test in amplitude damping environment for “X” state with density matrix &lt;inline-formula&gt;&lt;tex-math id="M11"&gt;\begin{document}${\rho _W}$\end{document}&lt;/tex-math&gt;&lt;alternatives&gt;&lt;graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="5-20222218_M11.jpg"/&gt;&lt;graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="5-20222218_M11.png"/&gt;&lt;/alternatives&gt;&lt;/inline-formula&gt;.