Influence of intermediated measurements on quantum statistical complexity of single driven qubit

Abstract

<sec> Recently, quantum statistical complexity based quantum information theory has received much attraction. Quantum measurements can extract the information from a system and may change its state. At the same time, the method of measuring multiple quantum is an important quantum control technique in quantum information science and condensed matter physics. The main goal of this work is to investigate the influence of multiple quantum measurements on quantum statistical complexity.</sec><sec> It is a fundamental problem to understand, characterize, and measure the complexity of a system. To address the issue, a damped and linearly driven two-level system (qubit) is taken for example. The driving amplitude and dephasing intensity are considered. By using the Lindblad equation and the Born-Markov approximation, the time evolution of the system can be obtained. Under multiple intermediated measurements, the system has a complex dynamic behavior. Quantum statistical complexity <inline-formula><tex-math id="M12">\begin{document}$C$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="23-20200802_M12.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="23-20200802_M12.png"/></alternatives></inline-formula> at the last moment <inline-formula><tex-math id="M13">\begin{document}$\tau$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="23-20200802_M13.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="23-20200802_M13.png"/></alternatives></inline-formula> is studied in detail. The results show that on the whole, <inline-formula><tex-math id="M14">\begin{document}$C$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="23-20200802_M14.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="23-20200802_M14.png"/></alternatives></inline-formula> first increases from zero to a maximal value with <inline-formula><tex-math id="M15">\begin{document}$\tau$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="23-20200802_M15.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="23-20200802_M15.png"/></alternatives></inline-formula> increasing, then decreases, and finally it approaches to zero. At first, the system is in a pure state and <inline-formula><tex-math id="M16">\begin{document}$C=0$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="23-20200802_M16.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="23-20200802_M16.png"/></alternatives></inline-formula>. Finally, the system is in a maximally mixed state due to the interaction with the environment and <inline-formula><tex-math id="M17">\begin{document}$C=0$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="23-20200802_M17.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="23-20200802_M17.png"/></alternatives></inline-formula> again. When the number of measurements <inline-formula><tex-math id="M18">\begin{document}$N$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="23-20200802_M18.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="23-20200802_M18.png"/></alternatives></inline-formula> is relatively small, <inline-formula><tex-math id="M19">\begin{document}$C$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="23-20200802_M19.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="23-20200802_M19.png"/></alternatives></inline-formula> fluctuates with <inline-formula><tex-math id="M20">\begin{document}$\tau$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="23-20200802_M20.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="23-20200802_M20.png"/></alternatives></inline-formula> increasing, but when <inline-formula><tex-math id="M21">\begin{document}$N$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="23-20200802_M21.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="23-20200802_M21.png"/></alternatives></inline-formula> is relatively large, the fluctuations disappear. Due to the quantum Zeno effect, as <inline-formula><tex-math id="M22">\begin{document}$N$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="23-20200802_M22.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="23-20200802_M22.png"/></alternatives></inline-formula> is larger, the variation of <inline-formula><tex-math id="M23">\begin{document}$C$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="23-20200802_M23.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="23-20200802_M23.png"/></alternatives></inline-formula> with <inline-formula><tex-math id="M24">\begin{document}$\tau$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="23-20200802_M24.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="23-20200802_M24.png"/></alternatives></inline-formula> is similar to that for the case of no intermediated measurement. Because of the quantum superposition principle, uncertainty principle, and quantum collapse, quantum measurement can disturb the system, so quantum statistical complexity <inline-formula><tex-math id="M25">\begin{document}$C$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="23-20200802_M25.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="23-20200802_M25.png"/></alternatives></inline-formula> exhibits a complex behavior.</sec><sec> In the quantum realm, the complexity of a system can be transformed into a resource. The quantum state needs creating, operating, or measuring. Therefore, all our results provide a theoretical reference for the optimal controlling of quantum information process and condensed matter physics. At the same time, the number of the degrees of freedom is two for the damped and linearly driven two-level system, so this system is simple and easy to study. The complexity of such a system can be tailored by properly tuning the driving strength. Therefore, the model can be used as a typical example to study the quantum statistical complexity.</sec>