Nice Error Basis and Quantum Channel

Abstract

The Weyl operators give a convenient basis of $M_n(ℂ)$ 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 $M_n(ℂ)$ to construct an NEB for $\mathrm{\text{Lin}}(M_n(ℂ))$, the space of linear maps on $M_n(ℂ)$. Any linear map will then correspond to a $n^2\times n^2$ coefficient matrix in the basis decomposition with respect to such an NEB of $\mathrm{\text{Lin}}(M_n(ℂ))$. Positivity, complete (co)positivity or other properties of a linear map can be characterised in terms of such a coefficient matrix.