Monotone Riemannian metrics and relative entropy on noncommutative probability spaces
Monotone Riemannian metrics and relative entropy on noncommutative probability spaces
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非交换概率空间上的单调黎曼度量和相对熵
DOI：
10.1063/1.533053
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发表时间：
1998
影响因子：
1.3
通讯作者：
M. Ruskai
中科院分区：
文献类型：
--
作者：
Andrew Lesniewski;M. Ruskai
利用相对模算子定义了（0，∞）上任意凸算子函数g的广义相对熵，g（1）=0.我们发现，这些凸算子函数可以划分成凸子集，每个子集定义了一个唯一的对称化相对熵，一个唯一的家庭（密度矩阵参数化）的连续单调黎曼度量，一个唯一的测地距离的空间密度矩阵，和一个唯一的单调算子函数满足一定的对称性和规范化条件。我们在几个重要的特殊情况下明确地描述了这些对象，包括g（w）=−log w，这产生了熟悉的对数相对熵。 相对熵，黎曼度量，测地距离通过我们的程序得到的合同下完全积极的，保迹映射。然后，我们定义和研究与这些量相关的最大收缩。
We use the relative modular operator to define a generalized relative entropy for any convex operator function g on (0,∞) satisfying g(1)=0. We show that these convex operator functions can be partitioned into convex subsets, each of which defines a unique symmetrized relative entropy, a unique family (parametrized by density matrices) of continuous monotone Riemannian metrics, a unique geodesic distance on the space of density matrices, and a unique monotone operator function satisfying certain symmetry and normalization conditions. We describe these objects explicitly in several important special cases, including g(w)=−log w, which yields the familiar logarithmic relative entropy. The relative entropies, Riemannian metrics, and geodesic distances obtained by our procedure all contract under completely positive, trace-preserving maps. We then define and study the maximal contraction associated with these quantities.