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expressing the LEP. Statement (ii) is the quantum analog of the inequality in Eq. (1). (The strict inequality is a minor mathematical detail; it arises from our insistence on using finite baths.) 

It has been shown in Theorem 1 that (i) implies (ii) and (iii). What is left is the proof of the implication (ii) ⇒ (i). Let Q be a hermitian operator such that J ( βQ ) > 0. A realization of the Landauer erasure will be constructed below and it will be shown that the associated HTO is equal to Q . 

a. The bath and its basis states. It is obvious that the bath needed for this purpose has infinitely many levels. Only in that case, it is possible to copy the information on A onto B and at the same time, retain the information already present in B. For better visualizing the information flow, we will think of the bath as being composed of infinitely many copies of a d -level system X. The constitution of B can be expressed as B = X$_{1}$X$_{2}$X$_{3}$ · · · . Let the standard orthonormal basis of X be denoted by {| i 〉} where i = 0 , 1 , . . . , d − 1. The following 

