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arXiv 1408.4477quant-ph
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Hellinger距离作为高斯不和谐的度量

Hellinger distance as a measure of Gaussian discord

Paulina Marian, Tudor A. Marian

摘要:

特别是近年来的研究,高斯不和谐可以量化为一个给定的双模高斯态和所有零不和谐的双模高斯态的集合之间的距离。然而,由于这个集合仅由乘积态组成,这样的距离捕获了模式之间的所有相关性(量子和经典)。因此,无论采用何种距离,它都只是几何不协调的一个上界。在这项工作中,我们选择了为此目的的海林格度规,已知有许多有益的属性推荐它作为一个很好的衡量量子行为。一般来说,这个度量是由亲和力决定的,这是一个与乌尔曼保真度相关的度量,它与乌尔曼保真度有许多重要的特征。作为我们工作的第一步,一对$n$模式高斯态的亲和力被写入。然后,在双模情况下,我们成功地确定了最接近的高斯乘积态,并计算了相应的高斯不和谐。得到的一般公式非常简单,在对称双模高斯态的重要情况下变得更加友好。然后,我们详细分析了两个特殊类的双模高斯态的理论和实验的兴趣,以及:压缩热态和模式混合热的。前者在一个众所周知的压缩阈值下是可分离的,而后者总是可分离的。值得强调的是,对于属于这两类的对称态,我们发现它们的几何Hellinger不和谐和高斯方法中最初定义的不和谐之间的一致性。与此同时,这种状态的高斯海灵格不和谐是其互相关总量的可靠度量。

英文摘要:

Especially investigated in recent years, the Gaussian discord can be quantified by a distance between a given two-mode Gaussian state and the set of all the zero-discord two-mode Gaussian states. However, as this set consists only of product states, such a distance captures all the correlations (quantum and classical) between modes. Therefore it is merely un upper bound for the geometric discord, no matter which is the employed distance. In this work we choose for this purpose the Hellinger metric that is known to have many beneficial properties recommending it as a good measure of quantum behaviour. In general, this metric is determined by affinity, a relative of the Uhlmann fidelity with which it shares many important features. As a first step of our work, the affinity of a pair of $n$-mode Gaussian states is written. Then, in the two-mode case, we succeeded in determining exactly the closest Gaussian product state and computed the Gaussian discord accordingly. The obtained general formula is remarkably simple and becomes still friendlier in the significant case of symmetric two-mode Gaussian states. We then analyze in detail two special classes of two-mode Gaussian states of theoretical and experimental interest as well: the squeezed thermal states and the mode-mixed thermal ones. The former are separable under a well-known threshold of squeezing, while the latter are always separable. It is worth stressing that for symmetric states belonging to either of these classes, we find consistency between their geometric Hellinger discord and the originally defined discord in the Gaussian approach. At the same time, the Gaussian Hellinger discord of such a state turns out to be a reliable measure of the total amount of its cross correlations.

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