Mathematical aspects of quantum statistical physics and quantum information theory

本年度的研究活动符合申请书中所述的项目主要目的。今年研究的主要成果是一个完整而简单的测地线凸迹函数集的表征。结果表明，由实函数的泛函演算给出的迹函数的测地线凸性与底层希尔伯特空间的维数无关。这个结果对于凸优化在不同的领域有广泛的应用，如机器学习、金融数学和量子物理中的多元算子映射。我们已经集中在最后的应用领域，并已经开发了一个相当一般的理论凸多元算子的意义。在多元算子的背景下，我们引入了所谓的熵超均值及其在量子信息论中的应用。我们提出了变形对数函数和指数函数的变分表示，并将其应用于量子Tsallis熵。最后，我们将Golden-Thompson的迹不等式推广到变形参数0<q<1的变形指数，从而补充了变形参数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.