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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. 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 $C$ at the last moment$\tau$ is studied in detail. The results show that on the whole,$C$ first increases from zero to a maximal value with$\tau$ increasing, then decreases, and finally it approaches to zero. At first, the system is in a pure state and$C=0$ . Finally, the system is in a maximally mixed state due to the interaction with the environment and$C=0$ again. When the number of measurements$N$ is relatively small,$C$ fluctuates with$\tau$ increasing, but when$N$ is relatively large, the fluctuations disappear. Due to the quantum Zeno effect, as$N$ is larger, the variation of$C$ with$\tau$ 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$C$ exhibits a complex behavior.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. -
Keywords:
- quantum statistical complexity /
- quantum measurement /
- quantum decoherence /
- a driven qubit
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图 1 中间测量次数不同时约化冯·诺依曼熵
$S$ 随最后演化时刻$\tau$ 的变化曲线 (a)$N=0, 1, 2, 4$ ; (b)$N=0, 10, 10^2, 10^3, 10^4$ , 外界驱动强度$\kappa=0.95$ , 退相位噪声强度$\gamma=0.2$ Figure 1. The reduced von Neumann entropy
$S$ varying with last moment$\tau$ , where (a)$N=0, 1, 2, 4$ , (b)$N=0, 10, 10^2, 10^3, 10^4$ , the driving amplitude$\kappa=0.95$ , the dephasing intensity$\gamma=0.2$ .
图 2 中间测量次数不同时约化失衡度D随最后演化时刻
$\tau$ 的变化曲线 (a)$N=0, 1, 2, 4$ ; (b)$N=0, 10, 10^2, 10^3, 10^4$ , 外界驱动强度$\kappa=0.95$ , 退相位噪声强度$\gamma=0.2$ Figure 2. The reduced disequilibrium
$D$ varying with last moment$\tau$ , where (a)$N=0, 1, 2, 4$ , (b)$N=0, 10, 10^2, 10^3, 10^4$ , the driving amplitude$\kappa=0.95$ , the dephasing intensity$\gamma=0.2$ .
图 3 中间测量次数不同时量子统计复杂度
$C$ 随最后演化时刻$\tau$ 的变化曲线 (a)$N=0, 1, 2, 4$ ; (b)$N=0, 10, 10^2, 10^3, 10^4$ , 外界驱动强度$\kappa=0.95$ , 退相位噪声强度$\gamma=0.2$ Figure 3. The quantum statistical complexity
$C$ varying with last moment$\tau$ , where (a)$N=0, 1, 2, 4$ , (b)$N=0, 10, 10^2, 10^3, 10^4$ , the driving amplitude$\kappa=0.95$ , the dephasing intensity$\gamma=0.2$ .
图 4 中间测量次数不同时量子统计复杂度
$C$ 随最后演化时刻$\tau$ 及外界驱动强度$\kappa$ 的变化 (a)$N=0$ ; (b)$N=4$ ; (c)$N=1000$ , 退相位噪声强度$\gamma=0.2$ Figure 4. The quantum statistical complexity
$C$ varying with last moment$\tau$ and driving amplitude$\kappa$ , where (a)$N=0$ , (b)$N=4$ , (b)$N=1000$ , and the dephasing intensity$\gamma=0.2$ . -
[1] Sen K D (Editor) 2011 Statistical Complexity: Applications in Electronic Structure (1st Ed.) (Netherlands: Springer) pp vii–xi
[2] 郝柏林 2001 物理 30 466
Google Scholar
Hao B L 2001 Physics 30 466
Google Scholar
[3] 冯端, 金国钧 2003 凝聚态物理学 (上卷) (北京: 高等教育出版社) 第4−8页
Feng D, Jin G J 2003 Condensed Matter Physics (Vol. 1) (Beijing: Higher Education Press) pp4−8 (in Chinese)
[4] Kadanoff L P 1991 Physics Today 44 9
[5] Boffetta G, Cencini M, Falcioni M, Vulpiani A 2002 Phys. Rep. 356 367
Google Scholar
[6] Kolmogorov A N 1965 Probl. Inform. Transm. 1 1
[7] Lempel A, Ziv J 1976 IEEE Trans. Inform. Theor. 22 75
Google Scholar
[8] López-Ruíz R, Mancini H L, Calbet X 1995 Phys. Lett. A 209 321
Google Scholar
[9] Shiner J S, Davison M, Landsberg P T 1999 Phys. Rev. E 59 1459
Google Scholar
[10] Cesário A T, Ferreira D L B, Debarba T, Iemini F, Maciel T O, Vianna R O 2020 https://arxiv.org/abs/2002.01590
[11] 解思深 2018 67 220301
Google Scholar
Xie S S 2018 Acta Phys. Sin. 67 220301
Google Scholar
[12] Friedenberger A, Lutz E 2017 Phys. Rev. A 95 022101
Google Scholar
[13] Friedenberger A, Lutz E 2018 https://arxiv.org/abs/1805.11882
[14] Bojer M, Friedenberger A, Lutz E 2019 J. Phys. Commun. 3 065003
Google Scholar
[15] 赵小新 2019 硕士学位论文 (南京: 南京邮电大学)
Zhao X X 2019 M. S. Dissertation (Nanjing: Nanjing University of Posts and Telecommunications) (in Chinese)
[16] Alter O, Yamamoto T, 2001 Quantum Measurement of a Single System (1st Ed.) (New York: Wiley Press) pp1−6
[17] Müller M M, Gherardini S, Smerzi A, Caruso F 2016 Phys. Rev. A 94 042322
Google Scholar
[18] Wiseman H M, Milburn G J 2009 Quantum measurement and control (1st Ed.) (Cambridge: Cambridge University Press) pp1−25
[19] Bernard D, Jin T, Shpielberg O 2018 Europhys. Lett. 121 60006
Google Scholar
[20] Misra B, Sudarshan E C 1977 J. Math. Phys. 18 756
Google Scholar
[21] 胡要花, 吴琴 2019 68 230303
Google Scholar
Hu Y H, Wu Q 2019 Acta Phys. Sin. 68 230303
Google Scholar
[22] Cai Y, Le H N, Scarani V 2015 Ann. Phys. 527 684
Google Scholar
[23] Ivanov D A, Gurvits L 2020 Phys. Rev. A 101 012303
Google Scholar
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