The Golden-Thompson trace inequality is complemented
The Golden-Thompson trace inequality is complemented
复制标题
Golden-Thompson 迹不等式得到补充
DOI：
10.1016/0024-3795(93)90029-n
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发表时间：
1993
影响因子：
1.1
通讯作者：
D. Petz
中科院分区：
文献类型：
--
作者：
F. Hiai;D. Petz
证明了一类迹不等式，它补充了Golden-Thompson不等式.例如，Tr（epA#epB）2/p ≠ TreA+ B对所有p> 0成立，其中A和B是埃尔米特矩阵，#表示几何平均值。我们还证明了涉及对数函数的相关迹不等式;即当X和Y是非负矩阵时，lp − 1 TrXlogYp/2XpYp/2 <$TrX（logX+logY）<$p− 1 TrXlogXp/2 YpXp/2（对于所有p> 0）。这些不等式提供了相对熵的上下界。
We prove a class of trace inequalities which complements the Golden-Thompson inequality. For example, Tr(epA#epB)2/p⩽ TreA+Bholds for allp> 0 whenAandBare Hermitian matrices and # denotes the geometric mean. We also prove related trace inequalities involving the logarithmic function; namelyp−1TrXlogYp/2XpYp/2⩽ TrX(logX+logY) ⩽p−1TrXlogXp/2YpXp/2for allp> 0 whenXandYare nonnegative matrices. These inequalities supply lower and upper bounds on the relative entropy.