The dynamics of open quantum systems are described using a set of operators called Kraus operators. In this paper, we show how to find system parameters for a closed system of two qubits undergoing quantum multiplexer operations such that Kraus operators can be written for a single open qubit system (the target qubit) when the initial density matrix of the joint system is in any arbitrary mixed state. The strategy used is to extend the single-qubit open system to a larger two-qubit closed system, which can evolve using unitary dynamics. The constructed two-qubit system is evolved using quantum multiplexer operations, and system parameters are derived to implement the operation. To derive the parameters, we use a reduced Hamiltonian technique wherein the qubit of interest (the target) evolves only in subspaces of the second qubit (the control). The main advantage of our scheme is that it is not restricted to separable product states and/or local unitary evolution of the joint system.