for all k . The flow of information is pictorially depicted in Fig. 1. Essentially the isometry shifts the information content of all subsystems of AB onto the next subsystem on the right. This flow is highly reminiscent of the motion of the guests in Hilbert’s Hotel Infinity paradox. In that hotel, all of the infinitely many rooms are occupied. But, it is still possible to accommodate a new guest by moving everyone to the next room. The isometry U$_{AB}$ essentially does the same for information. It retains the whole information already stored in B while it squeezes additional information coming from A into B. Our hotel, however, has an important difference from Hilbert’s hotel: as it follows from LEP, there is a nonzero check-in cost. In other words, while squeezing new information into B, the average energy of B increases. To be able to compute that increase, the bath Hamiltonian must be specified. c. The Hamiltonian of the bath. The Hamiltonian will be constructed such that each subsystem of the bath is independent and the energy eigenstates are identical with the standard basis states. The precise energy eigenvalues of the subsystem X$_{1}$ will be chosen to be identical with those of Q (plus a possible shift), i.e., H$_{X}$$_{1}$ = Q + c$_{1}$ 1$_{X}$$_{1}$ in the standard basis where here, c$_{1}$ is some constant. The energy eigenvalues of the remaining subsystems X$_{2}$, X$_{3}$, . . . will only shift the HTO by a multiple of identity. With this choice, the energy change of the bath is Δ E$_{B}$ = tr( ρ$_{A}$Q ) + c$_{2}$ where c$_{2}$ is a constant that is independent of ρ$_{A}$ . The value of c$_{2}$ depends on the energy levels of X$_{2}$, X$_{3}$, . . . . By adjusting these levels appropriately, it is possible set c$_{2}$ = 0. This shows that the HTO is Q and completes the proof. Since the details are complicated, the adjustment of the energy levels of other subsystems is discussed in Appendix A. □ The realization described in the proof above can be used for finding conditions for min- imizing the average heat emission during a Landauer erasure operation. Suppose that A is used as a memory device in an implementation of a quantum computer. Suppose also that A is subjected to a Landauer erasure at ambient temperature T$_{env}$ in some steps of the operation of the computer. Let ρ av A denote the average state of the device before the erasures are applied. The average heat emitted per erasure is then ¯ q = tr ρ av $_{A}$Q where Q is the HTO. The problem is to minimize ¯ q under the restriction that Q satisfies the conditions mentioned in Theorem 3. According to Theorem 1, ¯ q is bounded below by k$_{B}$T$_{env}$S ( ρ av $_{A}$). A simple inspection shows us that this bound is attained by a small error by the HTO Q = − k$_{B}$T$_{env}$ ln ρ av A + ϵ 1$_{A}$ (31) 17