Phrased differently, the duality is the isomorphism between completely positive maps (channels) from A to Cn×n, where A is a C*-algebra and Cn×n denotes the n×n complex entries, and positive linear functionals (states) on the tensor product Let H1 and H2 be (finite-dimensional) Hilbert spaces.
A quantum channel, in the Schrödinger picture, is a completely positive (CP for short), trace-preserving linear map that takes a state of system 1 to a state of system 2.
One can view ρΦ as a density matrix, and therefore the state dual to Φ.
The duality between channels and states refers to the map a linear bijection.
[2] More details on this can be found e.g. in the book Quantum Information Theory by M.