On the Matrix Representation of Quantum Operations
  This paper considers two frequently used matrix representations -- what we call the $\chi$- and $\mathcal{S}$-matrices -- of a quantum operation and their applications. The matrices defined with respect to an arbitrary operator basis, that is, the orthonormal basis for the space of linear operators on the state space are considered for a general operation acting on a single or two \textit{d}-level quantum system (qudit). We show that the two matrices are given by the expansion coefficients of the Liouville superoperator as well as the associated bijective, positive operator on the doubled-space defined with respect to two types of induced operator basis having different tensor product structures, i.e., Kronecker products of the relevant operator basis and dyadic products of the associated bipartite state basis. The explicit conversion formulas between the two matrices are established as a computable matrix multiplication. Extention to more qudits case is trivial. Several applications of these matrices and the conversion formulas in quantum information science and technology are presented. 
Yoshihiro Nambu and Kazuo Nakamura