Reduced superoperator method of solving the von Neumann equation

Abstract

The von Neumann equation, in superoperator representation, was solved for the qubit system. This required obtaining the eigenvalues and eigenoperators of the derivation superoperator. The basis operator method was proposed to compute for the eigenvalues. A new set of basis operators for the super-Hilbert space of the system was derived from the eigenoperators that correspond to zero eigenvalues (the solution of the homogeneous von Neumann equation) and the operators orthogonal to these. The clear advantage to this method, as opposed to directly evaluating the commutator, is the reduction of the degree of the characteristic polynomial related to the superoperator.