Contravariant densities, complete distancesand relative fidelities for quantum channels

Introducing contravariant trace densities for quantum states, we restore one-to-one correspondencebetween 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 is explicitly expressed in terms of these densities as the supremum over the input states. A larger C-distance is given as the natural norm distance for the channel densities, and another, Helinger type complete distance (CH distance), related to the minimax mean square fidelity optimization problem by purification of quantum channels, is also introduced and evaluated in terms of their contravariant trace densities. It is proved that the CH distance between two channels is equivalent to the CB distance. An operational meaning for these distances and relative complete fidelity for quantum channels is given in terms of quantum encodings producing optimal entanglements of quantum states for an opposite and output systems.