Symmetric states and dynamics of three quantum bits
Symmetric states and dynamics of three quantum bits
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DOI：
10.26421/qic22.7-8-1
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发表时间：
2021-11
期刊：
影响因子：
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通讯作者：
F. Albertini;D. D’Alessandro
中科院分区：
文献类型：
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作者：
F. Albertini;D. D’Alessandro
作用在希尔伯特空间上的幺正群${\cal H}：=（C^2）^{\otimes 3}$允许一个李群$U^{S_3}（8）$，其元素与三个对象的对称置换群进行置换。在这样一个李群的作用下，希尔伯特空间分裂为3个分别为4 $，2 $和2 $维的不变子空间，每个子空间对应于su（2）的一个不可约表示。维度为4 $的子空间是唯一确定的，并且对应于在对称群的作用下自身不变的状态。这就是所谓的对称扇区二维的子空间不是唯一确定的，我们将它们全部参数化。我们给出了在U^{S_3}（8）下不变的子空间中的纯态的分析。这涉及到它们的纠缠性质，可分性准则和李群$U^{S_3}（8）$下的动力学。作为我们研究的状态和动力学的物理动机，我们提出了一个物理设置，其中包括一个对称网络的三个自旋$\frac{1}{2}$粒子下一个共同的驱动电磁场。{For这样的系统，我们解决了驱动一个可分离态的状态与最大分布纠缠的控制理论问题。
The unitary group acting on the Hilbert space ${\cal H}:=(C^2)^{\otimes 3}$ of three quantum bits admits a Lie subgroup, $U^{S_3}(8)$, of elements which permute with the symmetric group of permutations of three objects. Under the action of such a Lie subgroup, the Hilbert space ${\cal H}$ splits into three invariant subspaces of dimensions $4$, $2$ and $2$ respectively, each corresponding to an irreducible representation of $su(2)$. The subspace of dimension $4$ is uniquely determined and corresponds to states that are themselves invariant under the action of the symmetric group. This is the so called {\it symmetric sector.} The subspaces of dimension two are not uniquely determined and we parametrize them all. We provide an analysis of pure states that are in the subspaces invariant under $U^{S_3}(8)$. This concerns their entanglement properties, separability criteria and dynamics under the Lie subgroup $U^{S_3}(8)$. As a physical motivation for the states and dynamics we study, we propose a physical set-up which consists of a symmetric network of three spin $\frac{1}{2}$ particles under a common driving electro-magnetic field. {For such system, we solve the control theoretic problem of driving a separable state to a state with maximal distributed entanglement.