FIG. 2: The isometry used in the realization for the complete erasure in the proof of the Theorem 2. The initial state in the top row is changed to the final state in the bottom row by copying the content of each subsystem to the next one on the right. for any allowed values of input labels. Essentially, the isometry U$_{AB}$ shifts the contents of the device and the subsystems in a chain which is infinite in two directions. The movement of the information among the subsystems is depicted in Fig. 2. Specifically, Note that U$_{AB}$ maps an orthonormal basis of AB to an orthonormal set and therefore it is an isometry. For the Hamiltonian, the subsystems are taken to be independent and the Hamiltonians of individual subsystems are described in terms of a continuous parameter. For this purpose, consider continuous functions ϵ$_{i}$ ( ν ) ( i = 0 , 1 , · · · , d − 1) and ϵ $^{′}$$_{j}$( ν ) ( j = 0 , 1 , · · · , r − 1) of a real parameter ν ∈ [0 , 1], to be used for energies of X and Y type subsystems respectively. They are chosen to satisfy the following properties. 29 (i) the content of Y$_{k}$ is copied to Y$_{k}$$_{−}$$_{1}$ for each k ≥ 2, (ii) the content of Y$_{1}$ is copied to A in such a way that the standard basis of Y$_{1}$ is carried onto the eigenvectors of ρ$_{0}$ , (iii) the content of A is copied to X$_{1}$ in such a way that the eigenvectors of Q is carried onto the standard basis of X$_{1}$, (iv) and the content of X$_{k}$ is copied onto X$_{k}$$_{+1}$ for all k . Y$_{1}$ Y$_{1}$ σ ′ 1 σ ′ 2 Y$_{2}$ Y$_{2}$ σ ′ 2 σ ′ 3 A A ρ ρ 0 X$_{1}$ X$_{1}$ σ 1 ρ X$_{2}$ X$_{2}$ σ 2 σ 1 X$_{3}$ X$_{3}$ σ 3 σ 2 Bath Bath