Tr A^p K^* B^（1-p）K的Lieb证明及有关不等式的注记
Lieb's simple proof of concavity of Tr A^p K^* B^(1-p) K and remarks on related inequalities
摘要：
给出了映射（A，B）--> Tr（A^pK ^* B^（1-p）K）的可解性的一个简单、完备的证明.作者没有声称原创性;这说明了利布的原始论点在其最简单的，而不是其最一般的形式。然后给出了从这个结果到A -> Tr e^[K + log（A）]的推论链的草图。给出了在熵不等式中起关键作用的映射（A，B，X）--> Tr \int X^*（A+ uI）^{-1} X（B+ uI）^{-1} du的联合凸性的一个独立的初等证明.
英文摘要：
A simple, self-contained proof is presented for the concavity of the map (A,B) --> Tr(A^p K^* B^(1-p) K). The author makes no claim to originality; this note gives Lieb's original argument in its simplest, rather than its most general, form. A sketch of the chain of implications from this result to concavity of A --> Tr e^[K + log(A)] is then presented. An independent elementary proof is given for the joint convexity of the map (A,B,X) --> Tr \int X^* (A+ uI)^{-1} X (B+ uI)^{-1} du which plays a key role in entropy inequalities.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
