In this paper we study the autoparallelity w.r.t. the e-connection for an
information-geometric structure called the SLD structure, which consists of a
Riemannian metric and mutually dual e- and m-connections, induced on the
manifold of strictly positive density operators. Unlike the classical
information geometry, the e-connection has non-vanishing torsion, which brings
various mathematical difficulties. The notion of e-autoparallel submanifolds is
regarded as a quantum version of exponential families in classical statistics,
which is known to be characterized as statistical models having efficient
estimators (unbiased estimators uniformly achieving the equality in the
Cramer-Rao inequality). As quantum extensions of this classical result, we
present two different forms of estimation-theoretical characterizations of the
e-autoparallel submanifolds. We also give several results on the
e-autoparallelity, some of which are valid for the autoparallelity w.r.t. an
affine connection in a more general geometrical situation