Ochoa Maicol A, Belzig Wolfgang, Nitzan Abraham
Department of Chemistry, University of Pennsylvania, Philadelphia, PA, 19104, USA.
Department of Physics, University of Konstanz, D-78457, Konstanz, Germany.
Sci Rep. 2018 Oct 25;8(1):15781. doi: 10.1038/s41598-018-33562-0.
In contrast to a projective quantum measurement, in a weak measurement the system is only weakly perturbed while only partial information on the measured observable is obtained. A simultaneous measurement of non-commuting observables cannot be projective, however the strongest possible such measurement can be defined as providing their values at the smallest uncertainty limit. Starting with the Arthurs and Kelly (AK) protocol for such measurement of position and momentum, we derive a systematic extension to a corresponding weak measurement along three steps: First, a plausible form of the weak measurement operator analogous to the Gaussian Kraus operator, often used to model a weak measurement of a single observable, is obtained by projecting a naïve extension (valid for commuting observable) onto the corresponding Gabor space. Second, we show that the so obtained set of measurement operators satisfies the normalization condition for the probability to obtain given values of the position and momentum in the weak measurement operation, namely that this set constitutes a positive operator valued measure (POVM) in the position-momentum space. Finally, we show that the so-obtained measurement operator corresponds to a generalization of the AK measurement protocol in which the initial detector wavefunctions is suitable broadened.
与投影量子测量不同，在弱测量中，系统仅受到微弱扰动，同时仅获取关于被测可观测量的部分信息。对非对易可观测量的同时测量不可能是投影测量，然而，最强可能的此类测量可定义为在最小不确定度极限下提供其值。从用于位置和动量此类测量的阿特斯和凯利（AK）协议出发，我们沿三个步骤推导出对相应弱测量的系统扩展：首先，通过将一个朴素扩展（对易可观测量有效）投影到相应的伽柏空间，得到一种类似于高斯克劳斯算子的弱测量算子的合理形式，高斯克劳斯算子常用于对单个可观测量的弱测量进行建模。其次，我们表明如此得到的测量算子集满足在弱测量操作中获得给定位置和动量值的概率的归一化条件，即该集在位置 - 动量空间中构成一个正算子值测度（POVM）。最后，我们表明如此得到的测量算子对应于AK测量协议的一种推广，其中初始探测器波函数得到适当展宽。
