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Entanglement types for two-qubit states with real amplitudes
Quantum Information Processing ( IF 2.5 ) Pub Date : 2021-03-03 , DOI: 10.1007/s11128-021-03025-z
Oscar Perdomo , Vicente Leyton-Ortega , Alejandro Perdomo-Ortiz

We study the set of two-qubit pure states with real amplitudes and their geometrical representation in the three-dimensional sphere. In this representation, we show that the maximally entangled states—those locally equivalent to the Bell states—form two disjoint circles perpendicular to each other. We also show that taking the natural Riemannian metric on the sphere, the set of states connected by local gates are equidistant to this pair of circles. Moreover, the unentangled or so-called product states are \(\pi /4\) units away to the maximally entangled states. This is, the unentangled states are the farthest away to the maximally entangled states. In this way, if we define two states to be equivalent if they are connected by local gates, we have that there are as many equivalent classes as points in the interval \([0,\pi /4]\) with the point 0 corresponding to the maximally entangled states. The point \(\pi /4\) corresponds to the unentangled states which geometrically are described by a torus. Finally, for every \(0< d < \pi /4\) the point d corresponds to a disjoint pair of torus. Finally, we also show how this geometrical interpretation allows to clearly see that any pair of two-qubit states with real amplitudes can be connected with a circuit that only has single-qubit gates and one controlled-Z gate.



中文翻译:

具有实际振幅的二量子位态的纠缠类型

我们研究了具有实际振幅的两量子位纯态集及其在三维球体中的几何表示。在此表示中,我们表明,最大纠缠状态(局部等效于贝尔状态)形成了两个彼此垂直的不相交的圆。我们还表明,采用球面上的自然黎曼度量,通过局部门连接的状态集与这对圆是等距的。此外,未纠缠或所谓的乘积状态为\(\ pi / 4 \)单位移到最大纠缠状态。这就是说,非纠缠态离最大纠缠态最远。这样,如果我们将两个状态定义为等效(如果它们通过本地门连接)是等效的,则我们可以得到区间\([[0,\ pi / 4] \)中点数为0的点的等效类数对应于最大纠缠态。点\(\ pi / 4 \)对应于在几何上由圆环描述的非纠缠状态。最后,对于每个\(0 <d <\ pi / 4 \),点d对应于不相交的一对圆环。最后,我们还展示了这种几何解释如何使您清楚地看到具有真实振幅的任何两对双量子位状态都可以与仅具有单量子位门和一个受控Z门的电路连接。

更新日期:2021-03-03
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