近日，德国锡根大学的Carlos de Gois和Matthias Kleinmann合作并取得一项新进展。经过不懈努力，他们实现用于量子态断层扫描的用户友好置信区间。相关研究成果已于2024年6月12日在国际知名学术期刊《物理评论A》上发表。

据悉，量子态层析成像是从实验数据重建量子态的标准技术。在有限统计的范围内，实验数据不能给出关于量子态的完美信息。表达这种有限知识的一种常用方法是在状态空间中提供置信区间。虽然以前提出了其他置信区间，但它们要么太浪费而不具有实际意义，要么不容易应用于一般测量方案，要么难以报道。

本文所构建的置信区间成功解决了这些问题，其优势在于渐近最优的样本成本和对实际参数的卓越性能。这些置信区间具有普适性，可应用于任何测量方案。它们还可由厄米算子空间中的椭球精确描述。该研究的构造基石是向量伯恩斯坦不等式，以及线性映射变换下的多项样本和的Hilbert-Schmidt范数误差的高概率界限。

附：英文原文

Title: User-friendly confidence regions for quantum state tomography

Author: Carlos de Gois, Matthias Kleinmann

Issue&Volume: 2024/06/12

Abstract: Quantum state tomography is the standard technique for reconstructing a quantum state from experimental data. In the regime of finite statistics, experimental data cannot give perfect information about the quantum state. A common way to express this limited knowledge is by providing confidence regions in the state space. Though other confidence regions were previously proposed, they are either too wasteful to be of practical interest, cannot easily be applied to general measurement schemes, or are too difficult to report. Here we construct confidence regions that solve these issues, as they have an asymptotically optimal sample cost and good performance for realistic parameters, are applicable to any measurement scheme, and can be described by an ellipsoid in the space of Hermitian operators. Our construction relies on a vector Bernstein inequality and bounds with high probability the Hilbert–Schmidt norm error of sums of multinomial samples transformed by linear maps.

DOI: 10.1103/PhysRevA.109.062417

Source: https://journals.aps.org/pra/abstract/10.1103/PhysRevA.109.062417