两个自旋耦合量子比特的精细纠缠和态操纵：Lie理论综述
Fine Entanglement and State Manipulation of Two Spin Coupled Qubits: A Lie Theoretic Overview
摘要：
通过建立在Kuzmak和Tkachuk的工作，“以施密特分解的形式制备两个自旋$\frac{1}{2}$粒子的量子态”，Physics Letters A，{ \bf 378}，pp 1469 -1474，其概述了该系统内纠缠度的控制，证明了，如果旋磁比为对于相等的旋磁比，不相等或三个方向。为了实现这一目标，一个基本的李理论证明的事实，即一组李代数向量的指数的有限产品所产生的变换是等于李群所产生的最小的李代数包含这些向量被改写成一个明确的算法。最后给出了一个数值算例，并证明了算法的有效性。
英文摘要：
By building on the work in Kuzmak & Tkachuk, "Preparation of quantum states of two spin-$\frac{1}{2}$ particles in the form of the Schmidt decomposition", Physics Letters A, {\bf 378}, pp1469-1474, which outlined the control of the degree of entanglement within this system, it is proven that any $SU(4)$ state manipulation operator can be realised for this system using a sequence of pulsed magnetic fields in either two linearly independent directions if the gyromagnetic ratios are unequal or three directions for equal gyromagnetic ratios. To achieve this goal, an elementary Lie theoretic proof of the fact that the group of transformations generated by finite products of exponentials of a set of Lie algebra vectors is equal to the Lie group generated by the smallest Lie algebra containing those vectors is rewritten into an explicit algorithm. A numerical example as well as the proof of the algorithm's effectiveness is given.
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