论文标题
$ n $级量子系统中的Bargmann不变性和弱价值的几何解释应用Majorana对称表示
Geometrical interpretation of the argument of Bargmann invariants and weak values in $N$-level quantum systems applying the Majorana symmetric representation
论文作者
论文摘要
在本文中，我们研究了一般可观察物的弱价值的论点，成功地对Bloch Sphere的这一论点进行了几何描述。我们应用Majoraana对称表示来实现此目标。一般可观察的弱价值与有效投影仪的弱价值成正比：它是由可观察到的在初始状态的标准化应用产生的，其相称性是真实的。 The argument of the weak value of a projector on a pure state of an $N$-level system corresponds to a symplectic area in the complex projective space $(\text{CP}^{N-1})$, which can be represented geometrically with a sum of $N-1$ solid angles on the Bloch sphere using the Majorana stellar representation.在这里，我们表明，可以使用Majorana表示，可以描述一般可观察到的弱价值的论点，是Bloch Sphere上$ N-1 $实心角的总和，并将两项研究融合在一起。这两种方法提供了两种几何描述，这是$ \ text {cp}^{n-1} $中的第一种方法，并在Bloch球上进行了第二种描述，在从原始空间$（\ text {cp}^{n-1} $中映射问题之后，通过使用Majoraana表示。这些结果也可以应用于三阶巴格曼不变式的参数，作为任何高阶不变式的参数，最基本的顺序可以表示为三阶巴格曼不变的参数的总和。最后，我们关注一般旋转1算子的弱价值的论点，当时它的模量向无穷大。这种差异在实验中具有很大有用的信号。
In this paper, we study the argument of weak values of general observables, succeeding to give a geometric description to this argument on the Bloch sphere. We apply the Majorana symmetric representation to reach this goal. The weak value of a general observable is proportional to the weak value of an effective projector: it arises from the normalized application of the observable over the initial state, with a constant of proportionality that is real. The argument of the weak value of a projector on a pure state of an $N$-level system corresponds to a symplectic area in the complex projective space $(\text{CP}^{N-1})$, which can be represented geometrically with a sum of $N-1$ solid angles on the Bloch sphere using the Majorana stellar representation. Here, we show that the argument of the weak value of a general observable can be described, using Majorana representation, as the sum of $N-1$ solid angles on the Bloch sphere, merging both studies. These two approaches provide two geometrical descriptions, a first one in $\text{CP}^{N-1}$ and a second one on the Bloch sphere, after mapping the problem from the original space $(\text{CP}^{N-1})$ by making use of the Majorana representation. These results can also be applied to the argument of the third-order Bargmann invariant, the most fundamental order as the argument of any higher order invariant can be expressed as a sum of the argument of third-order Bargmann invariants. Finally, we focus on the argument of the weak value of a general spin-1 operator when its modulus diverges towards infinity. This divergence amplifies signals with great usefulness in experiments.
