  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.
