Uncertainty relations for angular momentum
Uncertainty relations for angular momentum
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DOI：
10.1088/1367-2630/17/9/093046
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发表时间：
2015-09-25
影响因子：
3.3
通讯作者：
Werner, Reinhard F.
中科院分区：
文献类型：
--
作者：
Dammeier, Lars;Schwonnek, Rene;Werner, Reinhard F.
在这项工作中，我们研究了SU(2)的自旋s表示中角动量的各种不确定性概念。我们描述了由所有矢量给出的“不确定性区域”，其分量由三个角动量分量的方差指定。这个集合的一个基本特征是三个方差之和的下界。我们给出了一般算子三元组的不确定性区域的最优下界的一种方法，并对小s进行了计算。进一步的下界是通过推广Robertson获得状态相关下界的技术得到的。这些对于大s是最优的，因为它们被荷尔斯坦-普里马科夫近似的状态所饱和。我们证明，对于所有s，所有方差都符合所谓的向量模型，即它们也可以通过半径为根号s（s + 1）的球面上的经典概率测度来实现。熵不确定性关系也可以类似地讨论，但与最小化方差的状态不同，熵不确定性关系被最小化。对于大s， Maassen-Uffink界变得尖锐，我们明确地描述了极值状态。测量不确定性，正如最近由Busch， Lahti和Werner讨论的位置和动量，被引入，并明确确定了一个广义可观测值（POVM），它最小化了所有角动量分量的最坏情况测量不确定性，以及最小不确定性。最优测量的输出向量都具有相同的长度r(s)，其中r(s)/s -> 1等于s ->∞。
In this work we study various notions of uncertainty for angular momentum in the spin-s representation of SU(2). We characterize the 'uncertainty regions' given by all vectors, whose components are specified by the variances of the three angular momentum components. A basic feature of this set is a lower bound for the sum of the three variances. We give a method for obtaining optimal lower bounds for uncertainty regions for general operator triples, and evaluate these for small s. Further lower bounds are derived by generalizing the technique by which Robertson obtained his state-dependent lower bound. These are optimal for large s, since they are saturated by states taken from the Holstein-Primakoff approximation. We show that, for all s, all variances are consistent with the so-called vector model, i.e., they can also be realized by a classical probability measure on a sphere of radius root s(s + 1). Entropic uncertainty relations can be discussed similarly, but are minimized by different states than those minimizing the variances for small s. For large s the Maassen-Uffink bound becomes sharp and we explicitly describe the extremalizing states. Measurement uncertainty, as recently discussed by Busch, Lahti and Werner for position and momentum, is introduced and a generalized observable (POVM) which minimizes the worst case measurement uncertainty of all angular momentum components is explicitly determined, along with the minimal uncertainty. The output vectors for the optimal measurement all have the same length r(s), where r(s)/s -> 1 as s -> infinity.