  Quantum connections are defined by parallel transport operators acting on a
Hilbert space. They transport tangent operators along paths in parameter space.
The metric tensor of a Riemannian manifold is replaced by an inner product of
pairs of operator fields, similar to the inner product of the Kubo-Mori
formalism of Linear Response Theory. The metric is used to define the dual of a
quantum connection.The gradient of the parallel transport operators is the
quantum vector potential. It defines the covariant derivative of operator
fields. The covariant derivatives are used to quantify the holonomy of the
quantum connection. It is shown that a quantum connection is holonomic if and
only if its dual is holonomic. If the parallel transport operators are unitary
then an alpha-family of quantum connections can be defined in a way similar to
Amari's alpha family of connections in Information Geometry. The minus alpha
connection is the dual of the alpha connection. In particular, the alpha equal
zero connection is self-dual. An operator field can be combined with a path in
parameter space to produce a path in operator space. A definition is given for
such a path in operator space to be autoparallel. The path in parameter space
is then a geodesic for the induced connection.
