(25), then the density matrix that minimizes the expression (23), namely ρ$_{A}$ = σ$_{βQ}$ , gives equality in (14), and therefore by Theorem 1, ρ$_{A}$ must be isospectral with E ( ρ$_{A}$ ) = ρ$_{0}$ , i.e., the condition in (ii) holds. This completes the proof of the necessity of the conditions of Theorem 2. For the sufficiency proof, we have to start with a given Q that satisfies either (i) and (ii) and then show that a realization that yields E and Q can be constructed. The needed isometry is actually quite simple. It essentially copies the entire information on the device to some subsystem of the bath, so that the information is completely erased, and at the same time it copies the thermal equilibrium state of some subsystem of B to the device, so that the device has always the same final state. Even though the isometry is simple, the job of adjusting the HTO to be equal to the given operator Q is technically complicated. For this reason, the proof of the sufficiency of the conditions in Theorem 2 is done separately in Appendix B. However, it is appropriate to give a separate proof in here for the special case of Landauer erasures for which the required bath is simpler. Treatment of the technical details in this proof will also simplify the presentation of the full proof of Theorem 2 in Appendix B. Let E be a Landauer erasure to the standard state | ψ$_{0}$ 〉 ( E ( ρ$_{A}$ ) = | ψ$_{0}$ 〉 〈 ψ$_{0}$ | for all ρ$_{A}$ ). Note that the special case (ii) of Theorem 2 can hold only if the dimension of the Hilbert space of A is d = 1. This is a very trivial case where the system A cannot hold any information at all. In such a case, the HTO Q is a number and the theorem just says that this number is nonnegative, Q ≥ 0. For the generic case of d ≥ 2, however, only the strict inequality of (i) holds. The following is merely a corollary of Theorem 2 but we state it as a separate theorem and give a separate proof. Theorem 3 Let E be a Landauer erasure on a device A having d levels with d ≥ 2 and let Q be a hermitian operator. The following are equivalent. It is obvious that the statements (ii) and (iii) are equivalent; both are simply stating that J ( βQ ) is a strictly positive number. The statement (iii) is the conventional way of (i) Q is a possible HTO for a realization on a finite bath. (ii) tr e − βQ < 1 . (iii) tr ρ$_{A}$Q > k$_{B}$T S ( ρ$_{A}$ ) for all ρ$_{A}$ . 14