Mathematical aspects of quantum statistical physics and quantum information theory

量子统计物理和量子信息论的数学方面

• 批准号: 17K05267
• 负责人: HANSEN FRANK
• 金额: $ 2.83万
• 依托单位: Tohoku University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 2017
• 资助国家: 日本
• 起止时间: 2017-04-01 至 2019-03-31
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-17K05267/
• 关键词: geodesic convexity; quantum information; statistical mechanics; trace function; convexity; non-commutativity

This year’s research activities fall within the main purpose of the program as stated in the application.1. The main result in this year’s research is a complete and simple characterization of the set of geodesically convex trace functions. The result implies that geodesic convexity of a trace function given by the functional calculus of a real function is independent of the dimension of the underlying Hilbert space. This result has vast applications for convex optimization in diverse areas like machine learning, financial mathematics and multivariate operator maps with applications in quantum physics. We have focused on the last area of applications and have developed a quite general theory of convex multivariate operator means.2. In the context of multivariate operator means we have introduced the so-called entropic hyper-mean with applications in quantum information theory.3. We developed variational representations for the deformed logarithmic and exponential functions and applied them to the quantum Tsallis entropy. We finally extended Golden-Thompson’s trace inequality to deformed exponentials with deformation parameter 0<q<1, thus complementing earlier results for deformation parameter 1<q<2 and 2<q<3.

今年的研究活动属于申请书中所述的计划的主要目的。在今年的研究的主要成果是一个完整的和简单的特性的一套测地线凸迹函数。结果表明，由真实的函数的泛函微积分给出的迹函数的测地凸性与其所对应的Hilbert空间的维数无关。该结果在机器学习、金融数学和多元算子映射等不同领域的凸优化中具有广泛的应用，并在量子物理学中应用。我们重点研究了最后一个应用领域，并发展了一个相当一般的凸多元算子均值理论.在多元算子均值的背景下，我们引入了所谓的熵超均值，并将其应用于量子信息理论.我们开发的变形对数和指数函数的变分表示，并将其应用到量子Tsallis熵。最后，我们将Golden-Thompson迹不等式推广到变形参数为0<q<1的变形指数，从而补充了变形参数为1<q<2和2<q<3的早期结果.

项目成果

Convex multivariate operator means

凸多元算子均值

• DOI: 10.1016/j.laa.2018.11.032
• 发表时间: 2019
• 期刊: Linear Algebra and its Applications
• 影响因子: 1.1
• 作者: Hansen Frank

Variational representations related to Tsallis relative entropy

与 Tsallis 相对熵相关的变分表示

• 发表时间: 2019
• 期刊: arXiv: 1901.06807
• 作者: Hansen Frank;Hansen Frank and Guanghua Shi
• 通讯作者: Hansen Frank and Guanghua Shi

The entropic hyper-mean

熵超均值

• 发表时间: 2018
• 作者: Hansen Frank;Liang Jin;Shi Guanghua;Frank Hansen;Frank Hansen;Frank Hansen
• 通讯作者: Frank Hansen

Quantum Entropy Derived from First Principles

• DOI: 10.1007/s10955-016-1651-4
• 发表时间: 2016-11
• 期刊: Journal of Statistical Physics
• 影响因子: 1.6
• 作者: Frank Hansen

Operator maps of Jensen-type

Jensen型算子图

• DOI: 10.1007/s11117-018-0571-8
• 发表时间: 2018
• 期刊: Positivity
• 影响因子: 1
• 作者: Hansen Frank;Moslehian Mohammad Sal;Najafi Hamed
• 通讯作者: Najafi Hamed