量子散度是算子理论和量子信息论中基础且重要的研究课题，被广泛地用于量子信息、量子计算和统计学等领域，对量子纠缠和量子相干性等课题的研究具有十分重要的作用。本项目拟研究量子散度的协凸性及相关的算子不等式。具体包括：运用变分方法和复插值理论建立对称(反)范数下的Lieb凹性定理，研究某些量子散度的协凸性及其在量子信道或正保迹线性映射下的单调性，研究相关量子散度概念的推广；构造新的距离型量子散度，研究正定矩阵在该散度下的最小二乘问题并定义新的多元矩阵平均；运用算子凸性定理和majorization方法建立对称范数下的Golden-Thompson不等式和Peierls-Bogolyubov不等式，并研究量子散度的变分表达式和不等式关系。本项目的预期成果将为量子信息论中的度量问题提供重要的研究方法，并在算子方程、矩阵黎曼几何、最优传输理论和随机矩阵的研究中发挥重要作用。

The study of quantum divergences is a fundamental and significant research subject in operator theory and quantum information theory. Quantum divergences are widely used in fields of quantum information, quantum computation and statistics. Especially, they play an extremely important role in the research of quantum entanglement and quantum coherence. In this project we intend to study the joint convexity of quantum divergences and related operator inequalities. First of all, we will establish the generalized Lieb concavity theorem under the symmetric (anti-)norms by using of variational method and complex interpolation theory, construct the joint convexity of some quantum divergences and their monotonicity under quantum channel or positive trace preserving linear mappings, and generalize the concept of certain quantum divergences. Secondly, we will construct some new distance type quantum divergences, study the least square problem of positive definite matrices under them and then define some new multivariate matrix means. Last but not the least, we will establish the Golden-Thompson inequality and Peierls-Bogolyubov inequality under the symmetric norms by using of the operator convexity theorems and majorization method, and investigate the variational expression and inequalities of certain quantum divergences. The expected results of this project will not only provide some important mathematical methods for the study of measurement problems in quantum information theory, but also play an important role in the study of operator equation, matrix Riemannian geometry, optimal transform theory and random matrices.