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the current approach. Finally, Sec. V contains brief conclusions. 

## II. HEAT TRANSFER OPERATOR 

Let A be a quantum system with a finite number of levels which is used as a memory device for holding quantum information. A quantum operation on this system is a trace- preserving completely-positive map E that takes the initial state ρ$_{A}$ of the device to a final state E ( ρ$_{A}$ ). Although all quantum operations are within the scope of this article, special attention will be given to those that completely erase all information in the device. The operation E will be called a complete erasure if the final state is a constant mixed state ρ$_{0}$ for all initial states, i.e., 

<img src='content_image/334667.jpg'>

It will be called a Landauer erasure if it is a complete erasure with a pure final state, i.e., ρ$_{0}$ = | ψ$_{0}$ 〉 〈 ψ$_{0}$ | for some | ψ$_{0}$ 〉 . 

Any quantum operation E on A can be realized as a unitary transformation or an isometry U$_{AB}$ on a larger system AB, where B denotes an additional system that will henceforth be called as the bath. It is assumed that the device and the bath are initially uncorrelated. Hence, if the initial state of the bath is ρ$_{B}$ , the initial state of the composite system is the product state ρ$_{AB}$ = ρ$_{A}$ ⊗ ρ$_{B}$ . The final states after the application of the isometry will usually be denoted by primes, e.g., ρ ′ AB = U$_{AB}$ ( ρ$_{A}$ ⊗ ρ$_{B}$ ) U † $_{AB}$. We will say that the bath B, the state ρ$_{B}$ and the isometry U$_{AB}$ form a realization of the operation E if 

is satisfied for all ρ$_{A}$ , where here tr$_{B}$ denotes the partial trace over B. 

<img src='content_image/334666.jpg'>

It is assumed that the transformation U$_{AB}$ is obtained from the evolution of the composite system AB with a time-dependent Hamiltonian H$_{AB}$ ( t ). The basic assumption about the independence of the process (i.e., the Hamiltonian and therefore the transformation U$_{AB}$ ) from the information stored in the device is also made. In particular this means that no measurements are taken on the composite system AB during the process. Hence, during the transformation, any information exchange with A occurs entirely between A and B only and this process is described with the Hamiltonian H$_{AB}$ ( t ). No third system is involved in the transformation except in providing the energy for the necessary work done on AB. 

