Mathematical aspects of quantum statistical physics and quantum information theory

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演算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。这一结果在不同领域的凸优化中有着广泛的应用，如机器学习，金融数学和多元算子映射，以及量子物理学中的应用。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.

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.