Rank-based model selection for multiple ions quantum tomography
Rank-based model selection for multiple ions quantum tomography
复制标题
多离子量子层析成像的基于排序的模型选择
DOI：
10.1088/1367-2630/14/10/105002
复制
发表时间：
2012
影响因子：
3.3
通讯作者：
I. Dryden
中科院分区：
文献类型：
--
作者：
M. Guţă;T. Kypraios;I. Dryden
测量数据的统计分析已经成为许多量子工程实验的关键组成部分。由于标准的全态层析成像对于大维量子系统变得不可行，人们需要利用先验信息和实验态的稀疏性来降低估计问题的维度。在本文中，我们提出了模型选择的一般原则，通过对不同的模型进行拟合，并选择在似然拟合和模型复杂性之间具有最佳折衷的估计量，来寻找对数据的最简单或最简约的解释。我们应用了两种成熟的模型选择方法-Akaike信息准则(AIC)和贝叶斯信息准则(BIC)-这两个模型由固定秩状态和目前在多离子实验中产生的数据集组成。我们测试了AIC和BIC在随机选择的四个离子的低秩态上的性能，并研究了选择的秩数与一个离子态的测量重复次数之间的关系。然后，我们将这些方法应用于一个四离子实验的真实数据，该实验的目的是建立一个秩为4的斯莫林态。通过将这两种方法与皮尔逊χ2检验相结合，我们得出结论：该数据可以用一个秩介于7到9之间的模型来描述。此外，我们还发现，对于所有可能的测量，纯态的最大似然估计器的均方误差都接近于最优值。
The statistical analysis of measurement data has become a key component of many quantum engineering experiments. As standard full state tomography becomes unfeasible for large dimensional quantum systems, one needs to exploit prior information and the ‘sparsity’ properties of the experimental state in order to reduce the dimensionality of the estimation problem. In this paper we propose model selection as a general principle for finding the simplest, or most parsimonious explanation of the data, by fitting different models and choosing the estimator with the best trade-off between likelihood fit and model complexity. We apply two well established model selection methods—the Akaike information criterion (AIC) and the Bayesian information criterion (BIC)—two models consisting of states of fixed rank and datasets such as are currently produced in multiple ions experiments. We test the performance of AIC and BIC on randomly chosen low rank states of four ions, and study the dependence of the selected rank with the number of measurement repetitions for one ion states. We then apply the methods to real data from a four ions experiment aimed at creating a Smolin state of rank 4. By applying the two methods together with the Pearson χ2 test we conclude that the data can be suitably described with a model whose rank is between 7 and 9. Additionally we find that the mean square error of the maximum likelihood estimator for pure states is close to that of the optimal over all possible measurements.