  Introducing contravariant trace-densities for quantum states, we restore one
to one correspondence between quantum operations described by normal CP maps
and their trace densities as Hermitian positive operator-valued contravariant
kernels. The CB-norm distance between two quantum operations with type one
input algebras is explicitly expressed in terms of these densities, and this
formula is also extended to a generalized CB-distances between quantum
operations with type two inputs. A larger C-distance is given as the natural
norm-distance for the channel densities, and another, Helinger type distance,
related to minimax mean square optimization problem for purification of quantum
channels, is also introduced and evaluated in terms of their contravariant
trace-densities. It is proved that the Helinger type complete fidelity distance
between two channels is equivalent to the CB distance at least for type one
inputs, and this equivalence is also extended to type two for the generalized
CB distance. An operational meaning for these distances and relative complete
fidelity for quantum channels is given in terms of quantum encodings as
generalized entanglements of quantum states opposite to the inputs and the
output states.
