Mathematical aspects of quantum statistical physics and quantum information theory

今年的研究活动符合申请1中所述的计划的主要目的。今年的主要研究成果是对测地凸迹函数集进行了完整而简单的刻画。结果表明，由实函数的泛函演算给出的迹函数的测地凸性与下层Hilbert空间的维度无关。这一结果在机器学习、金融数学和多变量算子映射等不同领域的凸优化中有着广泛的应用，并在量子物理中得到了应用。我们已经把重点放在了最后一个应用领域，并发展了一个相当一般的多元凸算子平均理论。在多元算符方法的背景下，我们介绍了所谓的熵超平均及其在量子信息论中的应用。我们发展了变形的对数和指数函数的变分表示，并将它们应用于量子Tsallis熵。最后将Golden-Thompson迹不等式推广到形变参数为0&lt；q&lt；1的形变指数，补充了前人关于形变参数1&lt；q&lt；2和2&lt；q&lt；3的结果。

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.