In this addendum to our paper [ D. Burgarth and V. Giovannetti Phys. Rev. Lett. 99 100501 (2007)] we prove that during the transformation that allows one to enforce control by relaxation on a quantum system, the ancillary memory can be kept at a finite size, independently from the fidelity one wants to achieve. The result is obtained by introducing the quantum analog of defragmentation algorithms which are employed for efficiently reorganizing classical information in conventional hard disks.