Nice Error Basis and Quantum Channel

The Weyl operators give a convenient basis of Formula: see text which is also orthonormal with respect to the Hilbert-Schmidt inner product. The properties of such a basis can be generalised to the notion of a nice error basis (NEB), as introduced by E. Knill 3. We can use an NEB of Formula: see text to construct an NEB for Formula: see text, the space of linear maps on Formula: see text. Any linear map will then correspond to a Formula: see text coefficient matrix in the basis decomposition with respect to such an NEB of Formula: see text. Positivity, complete (co)positivity or other properties of a linear map can be characterised in terms of such a coefficient matrix.