在量子通訊上，糾纏被視為一寶貴資源，而糾纏即為各局域量子態之間的隱變量關係。吾人先考慮理想的兩粒子量子封閉系統，利用李雅普諾夫函數來控制量子態。然而吾人的控制重點與先前基於量子末態的控制方法不同，吾人是基於糾纏量測函數所獲得的新的李雅普諾夫函數，得以有效地控制量子態至最大糾纏態。此外吾人使用李雅普諾夫的第二種穩定性方法演示其李雅普諾夫穩定性。由於此種控制方法不仰賴量子最終狀態，於是更能有效地控制更高維或是更複雜的系統，能更有效的去進行控制。為何目前已有控制方法能確切的控制量子態至指定的末態，吾人卻還要設計一種無法決定量子末態的控制辦法？主要原因為在量子糾纏態的討論中，往往只需考慮其量子態之間的糾纏程度，而不一定要考慮到其最終狀態。在兩粒子系統中，對於糾纏程度較大的量子態，只要施以局域操作就可以製備實驗上所需的目標量子態，而不一定需要控制量子態至特定的最終狀態。面對更複雜的系統，實驗上想製備的是該系統中所能生成的最大糾纏態。但如何定義該系統處於最大糾纏態是相當困難的，也因此產生了無法確定最終狀態的問題。本研究論文的主要動機在提出一個控制方法，待找到量子系統的糾纏程度函數，即可控制該系統成為最大糾纏態。本論文以兩粒子的混合態為例，製備了最大糾纏混合態。目前已知的文獻並沒有對最大糾纏混合態有一決定性的數學式去描述，但吾人所提出的控制辦法可以製備最大糾纏混合態，這也間接解決了，目前缺乏最大糾纏混合態的通式的難題。本論文直接提供製備出最大糾纏混合態的辦法，此點不論在物理上亦或是控制領域上，都有巨大的貢獻。
找尋最大糾纏混合態的製備辦法，僅是吾人所提控制方法的其中一個應用，因為吾人所提出的控制辦法，更像是一種新的控制策略，它利用了量子理論的重要特性：量子系統之間的差別不在於末態，而在於糾纏的高低程度。

In quantum communication, entanglement is regarded as a precious resource, and entanglement is the hidden variable relationship between local quantum states. Firstly, we consider an ideal quantum closed system composed two particles, and use Lyapunov function to control the quantum state. However, our control methodology is different from the canonical control method, which is based on the quantum final state. We obtained a new Lyapunov function based on the entanglement measurement. It can effectively control the quantum state to the maximally entangled state. In addition, we use Lyapunov's second stability method to demonstrate its Lyapunov stability. Because our control method is independent to the final state. Hence, it can be more effectively applied to the more complex quantum system or the system with higher dimensions. Nowadays, there are some control methods already can accurately control the quantum state to the specific final state, why do we need to design another control method, which even do not consider the determination of the final state? The reason is that in a quantum state instead, only the degree of the entanglement is concerned. In a two-particles quantum system, the quantum state with a greater degree of entanglement can be used to prepare the experimentally required target quantum state as long as it is operated locally. In other words, when preparing the entangled state at the beginning, it is not necessary to control the quantum state to a specific final state, as long as the quantum state has a greater degree of entanglement than the final state. Then, we can prepare the final state locally. In a more complex system, the experimental desired quantum state is undoubtedly the maximally entangled state, however, it is very difficult to determine. Therefore, a problem arises with this difficulty, that is the undetermined final state. Using the control method proposed in this paper, as long as the function that determines the degree of entanglement of the system can be found, the quantum system can be controlled to the maximally entangled state. This paper takes the mixed state of two particles as an example to prepare the maximally entangled mixed state. The currently known literature does not have a decisive mathematical formula to describe the maximally entangled mixed state. However, our proposed control method can prepare the maximally entangled mixed state, which also provides an indirect solution for the missing mathematical description of the maximally entangled mixed state. problem. Our result directly contributes to the method of preparing the maximal entangled mixed state, which has a huge contribution no matter in physics or in the field of control.
Finding the method of preparing the maximally entangled mixed state is only one of the applications of the control method proposed by this thesis, because the control method proposed here is more like a new control strategy, which takes advantage of the important characteristic of the quantum theory: the difference between two quantum system relays on the degree of the entanglement, not the final state.

[1]P. I. Hagouel and I. G. Karafyllidis, “Quantum computers: Registers, gates and algorithms,”2012 28thInternational Conference on Microelectronics Proceedings, 05 2012.
[2]W.K. Wootters, “Entanglement of formation of an arbitrary state of two qubits,”Physical Review Letters,vol. 80, pp. 2245–2248, 03 1998.[3]V.S.MalinovskyandI.R.Sola, Quantumcontrolofentanglementbyphasemanipulationoftime­delayedpulse sequences. i,”Physical Review A, vol. 70, 10 2004.
[4]A. Iserles, H. Z. Munthe­Kaas, S. P. Nørsett, and A. Zanna, “Lie­group methods,”Acta Numerica, vol. 9,pp. 215–365, 01 2000.
[5]Botsinis, Panagiotis, S. X. Ng, and L. Hanzo, “Quantum search algorithms, quantum wireless, and a low­complexity maximum likelihood iterative quantum multi­user detector design,”IEEE Access, vol. 1, pp.94–122, 2013.
[6]R.Nevels,J.Jeong,andP.Hemmer,“Microwavesimulationofgrover’squantumsearchalgorithm,”IEEEAntennas and Propagation Magazine, vol. 48, pp. 38–47, 10 2006.
[7]J. Gough, V. Belavkin, and O. Smolyanov, “Hamilton­jacobi­bellman equations for quantum optimalcontrol,”DAYS on DIFFRACTION 2006, 2006.
[8]Q.­F. Wang, “Quantum optimal control of nonlinear dynamics systems described by klein­gordon­schrodinger equations,”2006 American Control Conference, 2006.
[9]——, “Quantum optimal control of nuclei in the presence of perturbation in electric field,”IET ControlTheory Applications, vol. 3, pp. 1175–1182, 09 2009.[10]R. Schmidt, A. Negretti, J. Ankerhold, T. Calarco, and J. T. Stockburger, “Optimal control of open quan­tum systems: Cooperative effects of driving and dissipation,”Physical Review Letters, vol. 107, 09 2011.
[11]M. Hintermüller, D. Marahrens, P. A. Markowich, and C. Sparber, “Optimal bilinear control of gross–pitaevskii equations,”SIAM Journal on Control and Optimization, vol. 51, pp. 2509–2543, 01 2013.
[12]J.­S. Li, J. Ruths, T.­Y. Yu, H. Arthanari, and G. Wagner, “Optimal pulse design in quantum control: Aunified computational method,”Proceedings of the National Academy of Sciences, vol. 108, pp. 1879–1884, 01 2011.
[13]D.DongandP.I.R,“Sampled­datacontroloftwo­levelquantumsystemsbasedonslidingmodedesign,”50th IEEE Conference on Decision and Control and European Control Conference (CDC­ECC),Orlando,FL, USA, p. 6236–6241, 12 2011.
[14]K. Mishima and K. Yamashita, “Free­time and fixed end­point optimal control theory in quantum me­chanics: Application to entanglement generation,”The Journal of Chemical Physics, vol. 130, p. 034108,01 2009.
[15]U. V. Boscain, F. Chittaro, P. Mason, and M. Sigalotti, “Adiabatic control of the schrödinger equation viaconical intersections of the eigenvalues,”IEEE Transactions on Automatic Control, vol. 57, pp. 1970–1983, 08 2012.
[16]N.Smaoui,A.El­Kadri,andM.Zribi,“Adaptiveboundarycontroloftheunforcedgeneralizedkorteweg–de vries–burgers equation,”Nonlinear Dynamics, vol. 69, pp. 1237–1253, 02 2012.
[17]S. H. B, S. , and P. Rouchon, “A time­periodic lyapunov approach for motion planning of controllabledriftless systems on su(n),”Proceedings of the 48h IEEE Conference on Decision and Control (CDC)held jointly with 2009 28th Chinese Control Conference, 12 2009.
[18]Q.Zhang,W.Wang,andL.Wang,“Quantumsystemcontrolbasedonlyapunovtechnology,”2008FourthInternational Conference on Natural Computation, 2008.[19]U.Boscain,F.C.Chittaro,P.Mason,R.Pacqueau,andM.Sigalotti,“Motionplanninginquantumcontrolvia intersection of eigenvalues,”49th IEEE Conference on Decision and Control (CDC), 12 2010.
[20]S. Kuang and S. Cong, “Lyapunov control methods of closed quantum systems,”Automatica, vol. 44, pp.98–108, 01 2008.
[21]D. Dong and I. R. Petersen, “Sliding mode control of two­level quantum systems,”Automatica, vol. 48,pp. 725–735, 05 2012.
[22]X. X. Yi, S. L. Wu, C. Wu, X. L. Feng, and C. H. Oh, “Time­delay effects and simplified control fieldsin quantum lyapunov control,”Journal of Physics B: Atomic, Molecular and Optical Physics, vol. 44, p.195503, 09 2011.
[23]J. WEN and S. CONG, “Transfer from arbitrary pure state to target mixed state for quantum systems,”IFAC Proceedings Volumes, vol. 44, pp. 4638–4643, 01 2011.
[24]M. Mirrahimi, P. Rouchon, and G. Turinici, “Lyapunov control of bilinear schrödinger equations,”Auto­matica, vol. 41, pp. 1987–1994, 11 2005.
[25]S. Kuang, S. Cong, and Y. Lou, “Population control of quantum states based on invariant subsetsunder a diagonal lyapunov function,”IEEE Xplore, p. 2486–2491, 12 2009. [Online]. Available:https://ieeexplore.ieee.org/document/5399699
[26]S. Grivopoulos and B. Bamieh, “Lyapunov­based control of quantum systems,”42nd IEEE InternationalConference on Decision and Control (IEEE Cat. No.03CH37475)
[27]F. Yang and S. Cong, “Preparation of entanglement states in a two­spin system by lyapunov­basedmethod,”Journal of Systems Science and Complexity, vol. 25, pp. 451–462, 06 2012.
[28]X. Wangand S. Schirmer, “Analysis of lyapunov control for hamiltonian quantum systems,”Proceedingsof the ENOC, Saint Petersburg, Russia, 05 2008.
[29]X. Wang and S. G. Schirmer, “Analysis of lyapunov method for control of quantum states,”IEEE Trans­actions on Automatic Control, vol. 55, pp. 2259–2270, 10 2010.
[30]——, “Analysisofeffectivenessoflyapunovcontrolfornon­genericquantumstates,”IEEE Transactionson Automatic Control, vol. 55, pp. 1406–1411, 06 2010.
[31]Q. Fan, “Generation of bell states via lyapunov control on a two­qubit system with an anisotropic xyheisenberg interaction,”Science China Physics, Mechanics and Astronomy, vol. 54, pp. 474–478, 022011.
[32]F. Yang and S. Cong, “Purification of mixed state for two­dimensional systems via interaction control,”IEEE Xplore, vol. 2, p. 91–94, 10 2010. [Online]. Available: https://ieeexplore.ieee.org/document/5743387
[33]J. Liu and S. Cong, “Trajectory tracking of quantum states based on lyapunov method,”2011 9th IEEEInternational Conference on Control and Automation (ICCA), 12 2011.
[34]W. Yang and J. Sun, “One lyapunov control for quantum systems and its application to entanglementgeneration,”Physics Letters A, vol. 377, pp. 851–854, 05 2013.
[35]S. Cong and F. Yang, “Control of quantum states in decoherence­free subspaces,”Journal of Physics A:Mathematical and Theoretical, vol. 46, p. 075305, 02 2013.
[36]Y. Lou, J. Yang, S. Kuang, and S. Cong, “Path programming control strategy of quantum state transfer,”IET Control Theory Applications, vol. 5, pp. 291–298, 01 2011.
[37]S. KUANG and S. CONG, “Population control of equilibrium states of quantum systems via lyapunovmethod,”Acta Automatica Sinica, vol. 36, pp. 1257–1263, 12 2010.
[38]K. Beauchard, J. M. Coron, M. Mirrahimi, and P. Rouchon, “Implicit lyapunov control of finite dimen­sional schrödinger equations,”Systems Control Letters, vol. 56, pp. 388–395, 05 2007.
[39]A. Grigoriu, “Implicit lyapunov control for schrödinger equations with dipole and polarizability term,”IEEE Conference on Decision and Control and European Control Conference, 12 2011.
[40]S. Zhao, H. Lin, J. Sun, and Z. Xue, “An implicit lyapunov control for finite­dimensional closed quantumsystems,”International Journal of Robust and Nonlinear Control, vol. 22, pp. 1212–1228, 05 2011.
[41]F. Meng, S. Cong, and S. Kuang, “Implicit lyapunov control of multi­control hamiltonian systems basedon state distance,”Proceedings of the 10th World Congress on Intelligent Control and Automation, 072012.
[42]J. Sharifi and H. Momeni, “Lyapunov control of squeezed noise quantum trajectory,”Physics Letters A,vol. 375, pp. 522–528, 01 2011.
[43]F. Ticozzi, R. Lucchese, P. Cappellaro, and L. Viola, “Hamiltonian control of quantum dynamical semi­groups: Stabilization and convergence speed,”IEEE Transactions on Automatic Control, vol. 57, pp.1931–1944, 08 2012.
[44]M. Nielsen and I. Chuang,Quantum Computation and Quantum Information. Cambridge UniversityPress, 2000. [Online]. Available: http://csis.pace.edu/~ctappert/cs837­19spring/QC­textbook.pdf
[45]N. J. Cerf and C. Adami, “Information theory of quantum entanglement and measurement,”Physica D:Nonlinear Phenomena, vol. 120, pp. 62–81, 09 1998.[46]C. Xu, “Completely positive matrices,”Linear Algebra and its Applications, vol. 379, pp. 319–327, 032004.
[47]A.Younes, J.Rowe, andJ.Miller, “Enhancedquantumsearchingviaentanglementandpartialdiffusion,”Physica D: Nonlinear Phenomena, vol. 237, pp. 1074–1078, 06 2008.
[48]M.A.Nielsen,“Conditionsforaclassofentanglementtransformations,”PhysicalReviewLetters,vol.83,pp. 436–439, 07 1999.
[49]R. F. Werner, “Quantum states with einstein­podolsky­rosen correlations admitting a hidden­variablemodel,”Physical Review A, vol. 40, pp. 4277–4281, 10 1989.
[50]M. Horodecki, P. Horodecki, and R. Horodecki, “Separability of mixed states: necessary andsufficient conditions,”Physics Letters A, vol. 223, pp. 1–8, 11 1996. [Online]. Available:https://arxiv.org/abs/quant­ph/9605038
[51]B. M. Terhal, “Bell inequalities and the separability criterion,”Physics Letters A, vol. 271, pp. 319–326,07 2000.
[52]A. Peres, “Separability criterion for density matrices,”Physical Review Letters, vol. 77, pp. 1413–1415,08 1996. [Online]. Available: https://arxiv.org/abs/quant­ph/9604005
[53]N.Gisin,“Hiddenquantumnonlocalityrevealedbylocalfilters,”PhysicsLettersA,vol.210,pp.151–156,01 1996.
[54]S. Rana, “Negative eigenvalues of partial transposition of arbitrary bipartite states,”Physical Review A,vol. 87, 05 2013.
[55]E. Størmer, “Positive linear maps of operator algebras,”Acta Mathematica, vol. 110, pp. 233–278, 1963.
[56]S. Woronowicz, “Positive maps of low dimensional matrix algebras,”Reports on Mathematical Physics,vol. 10, pp. 165–183, 10 1976.
[57]M. Plenio and S. Virmani, “An introduction to entanglement measures,”Quantum Information and Com­putation, vol. 7, pp. 1–51, 01 2007.
[58]G. Vidal, “Entanglement monotones,”Journal of Modern Optics, vol. 47, pp. 355–376, 02 2000.
[59]A. Uhlmann, “Entropy and optimal decompositions of states relative to a maximal commutative subalge­bra,”Open Systems Information Dynamics, vol. 5, pp. 209–228, 1998.
[60]R. Horodecki, P. Horodecki, M. Horodecki, and K. Horodecki, “Quantum entanglement,”Reviews ofModern Physics, vol. 81, pp. 865–942, 06 2009.
[61]J. Preskill, “Lecture notes for physics 219: Quantum computation,” 01 1999.[62]E.Herterich,“Suddendeathofentanglementfornon­localityandconcurrence : Areviewofentanglementsudden death behaviour of non­locality and concurrence in commonly used entangled state classes underinuence of decay and dephasing noise dynamics,” 2019.
[63]S.AlbeverioandS.­M.Fei, “Anoteoninvariantsandentanglements,”Journal of Optics B: Quantum andSemiclassical Optics, vol. 3, pp. 223–227, 07 2001.
[64]M. Li, S.­M. Fei, and Z.­X. Wang, “A lower bound of concurrence for multipartite quantum states,”Journal of Physics A: Mathematical and Theoretical, vol. 42, p. 145303, 03 2009.
[65]Z.­H. Ma, Z.­H. Chen, J.­L. Chen, C. Spengler, A. Gabriel, and M. Huber, “Measure of genuine multi­partite entanglement with computable lower bounds,”Physical Review A, vol. 83, 06 2011.
[66]Z.­H. Chen, Z.­H. Ma, J.­L. Chen, and S. Severini, “Improved lower bounds on genuine­multipartite­entanglement concurrence,”Physical Review A, vol. 85, 06 2012.
[67]F. Mintert, M. Kuś, and A. Buchleitner, “Concurrence of mixed multipartite quantum states,”PhysicalReview Letters, vol. 95, 12 2005.
[68]S. Ishizaka and T. Hiroshima, “Maximally entangled mixed states under nonlocal unitary operations intwo qubits,”Physical Review A, vol. 62, 07 2000.[69]M. Christandl and A. Winter, ““squashed entanglement”: An additive entanglement measure,”Journalof Mathematical Physics, vol. 45, pp. 829–840, 03 2004.
[70]K. Chen, S. Albeverio, and S.­M. Fei, “Entanglement of formation of bipartite quantum states,”PhysicalReview Letters, vol. 95, 11 2005.
[71]T. J. Osborne, “Entanglement measure for rank­2 mixed states,”Physical Review A, vol. 72, 08 2005.
[72]R.Demkowicz­Dobrzański,A.Buchleitner,M.Kuś,andF.Mintert,“Evaluablemultipartiteentanglementmeasures: Multipartite concurrences as entanglement monotones,”Physical Review A, vol. 74, 11 2006.
[73]C.Eltschka,T.Bastin,A.Osterloh,andJ.Siewert,“Multipartite­entanglementmonotonesandpolynomialinvariants,”Physical Review A, vol. 85, 02 2012.
[74]Y. Most, Y. Shimoni, and O. Biham, “Formation of multipartite entanglement using random quantumgates,”Physical Review A, vol. 76, 08 2007.
[75]A. Osterloh and J. Siewert, “Constructingn­qubit entanglement monotones from antilinear operators,”Physical Review A, vol. 72, 07 2005