论文标题
操作员缩放的信息几何形状
Information geometry of operator scaling
论文作者
论文摘要
矩阵缩放是广泛应用的经典问题。众所周知，从经典信息几何学的角度来看，用于矩阵缩放的sindhorn算法被解释为交替的电子投影。最近，发现矩阵缩放对完全积极的地图的概括（称为运算符缩放）出现在数学和计算机科学的各个领域中，并且Sinkhorn算法已扩展到运算符缩放器。在这项研究中，从量子信息几何形状的角度研究了操作员sindhorn算法，该算法是通过完全正面地图的Choi表示。表明操作员sindhorn算法与相对于对称对数衍生品度量的交替电子投影相吻合，这是与量子估计理论相关的量子状态空间上的riemannian公制。还通过在正定锥上使用不同信息几何结构来提供其他类型的交流电子投影算法。
Matrix scaling is a classical problem with a wide range of applications. It is known that the Sinkhorn algorithm for matrix scaling is interpreted as alternating e-projections from the viewpoint of classical information geometry. Recently, a generalization of matrix scaling to completely positive maps called operator scaling has been found to appear in various fields of mathematics and computer science, and the Sinkhorn algorithm has been extended to operator scaling. In this study, the operator Sinkhorn algorithm is studied from the viewpoint of quantum information geometry through the Choi representation of completely positive maps. The operator Sinkhorn algorithm is shown to coincide with alternating e-projections with respect to the symmetric logarithmic derivative metric, which is a Riemannian metric on the space of quantum states relevant to quantum estimation theory. Other types of alternating e-projections algorithms are also provided by using different information geometric structures on the positive definite cone.
