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arXiv 2003.14390quant-ph
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三量子比特系统中纠缠的矢量性质

Vector Properties of Entanglement in a Three-Qubit System

Dmitry B. Uskov, Paul M. Alsing

备注：
26 pages, no figures; Accepted Phys. Rev. A 29Jul2020
发表信息：
Phys. Rev. A 102, 032401 (2020)
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摘要：

基于$su(4)$和$so(6)$李代数之间的同构，提出了一个三量子比特系统中纠缠的动力学矢量模型.推广Plücker型描述的三个量子比特的本地不变量，我们引入了三对实值的$3D$矢量（这里表示为$A_{R,I}$，$B_{R,I}$和$C_{R,I}$）。这些矢量的大小决定了系统的两量子比特和三量子比特纠缠参数。我们证明了在局域$SU(2)$操作下矢量$A$，$B$，$C$的演化与相应的单比特Bloch矢量$a$，$b$，$c$的$SO(3)$演化是一致的.同时，包含$a-b$、$a-c$和$b-c$两量子位耦合项的一般两量子位$su(4)$哈密顿算子相应地产生矢量$A$和$B$、$A$和$C$以及$B$和$C$之间的$SO(6)$耦合。结果表明，不同的两量子比特耦合项所引起的纠缠动力学完全取决于矢量$A$，$B$，$C$的相互取向，而这些取向可以通过单量子比特变换来控制。我们通过解决涉及$W$，Greenberg-Horne-Zeilinger（$GHZ$）和biparseparable状态之间的转换的量子控制问题来说明这种矢量描述的纠缠的力量。

英文摘要：

We suggest a dynamical vector model of entanglement in a three qubit system based on isomorphism between $su(4)$ and $so(6)$ Lie algebras. Generalizing Plücker-type description of three-qubit local invariants we introduce three pairs of real-valued $3D$ vector (denoted here as $A_{R,I}$ , $B_{R,I}$ and $C_{R,I}$). Magnitudes of these vectors determine two- and three-qubit entanglement parameters of the system. We show that evolution of vectors $A$, $B$ , $C$ under local $SU(2)$ operations is identical to $SO(3)$ evolution of single-qubit Bloch vectors of qubits $a$, $b$ and $c$ correspondingly. At the same time, general two-qubit $su(4)$ Hamiltonians incorporating $a-b$, $a-c$ and $b-c$ two-qubit coupling terms generate $SO(6)$ coupling between vectors $A$ and $B$, $A$ and $C$, and $B$ and $C$, correspondingly. It turns out that dynamics of entanglement induced by different two-qubit coupling terms is entirely determined by mutual orientation of vectors $A$, $B$, $C$ which can be controlled by single-qubit transformations. We illustrate the power of this vector description of entanglement by solving quantum control problems involving transformations between $W$, Greenberg-Horne-Zeilinger ($GHZ$ ) and biseparable states.

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