Monotone Riemannian metrics on density matrices with non-monotone scalar curvature
Monotone Riemannian metrics on density matrices with non-monotone scalar curvature
复制标题
具有非单调标量曲率的密度矩阵的单调黎曼度量
DOI：
10.1063/1.1592874
复制
发表时间：
2003
期刊：
影响因子：
--
通讯作者：
A. Andai
中科院分区：
文献类型：
--
作者：
A. Andai
量子系统状态空间上的单调黎曼度量理论是由Denes Petz于1996年建立的。在最近的一篇论文中，他认为统计相关单调度量的标量曲率可以解释为平均统计不确定性。本文件有助于这一主题。可以合理地预期，混合程度较高的状态比混合程度较低的状态更难区分。这种行为的表现可能是，对于这样的度量，标量曲率在最大混合状态下具有最大值。我们发现，并不是每一个单调度量满足这一期望，其中一些表现在一个非常不同的方式。给出了单调黎曼度量在最大混合状态下具有局部极小值的数学条件，并给出了这类度量的例子。
The theory of monotone Riemannian metrics on the state space of a quantum system was established by Denes Petz in 1996. In a recent paper he argued that the scalar curvature of a statistically relevant—monotone—metric can be interpreted as an average statistical uncertainty. The present paper contributes to this subject. It is reasonable to expect that states which are more mixed are less distinguishable than those which are less mixed. The manifestation of this behavior could be that for such a metric the scalar curvature has a maximum at the maximally mixed state. We show that not every monotone metric fulfils this expectation, some of them behave in a very different way. A mathematical condition is given for monotone Riemannian metrics to have a local minimum at the maximally mixed state and examples are given for such metrics.