 We derive quantum non-equilibrium equalities for absolutely irreversible processes, where the density matrix does not return to the subspace spanned by initial eigenvectors, and discuss a feedback control model on a Corbett system to illustrate these equalities. We show how entropy reduction via feedback control can be converted into work in the presence of absolute irreversibility, and provide an explicit form for extractable work. This article was authored by Ken Funo, Yuto Murashita, and Mossajito Ueda.