Nice Error Basis and Quantum Channel

Abstract

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.