content_image/372506.jpg content_image/372507.jpg content_image/372508.jpg
where ϵ is a small positive number. The construction in the proof of Theorem 3 can then be followed for adjusting the energy levels of the bath, the detailed Hamiltonian H$_{AB}$ ( t ) during device-bath interaction and the consequent isometry. In a realistic situation where the environment serves as the bath B, it is not possible to modify the energy levels of B at will and the Landauer bound may not be attained; but the construction above can still be useful. For example, a finite chain X$_{1}$X$_{2}$ · · · X$_{N}$ can be used for connecting the device and the environment. By adjusting the energy levels of the subsystems in the chain and using the same information flow, ¯ q can still be significantly decreased. 

## C. General Quantum Operations 

A generic quantum operation causes only a partial erasure of the quantum information stored by the device by forming an entanglement between the device and the bath during the realization. Even though the inequality (14) continues to be valid for the generic case, it does not necessarily give a complete characterization of the possible HTOs as it will be discussed in Sec. V. This subsection contains a few basic results about the HTOs of generic quantum operations that surpass (14) in implications. 

Let E be a quantum operation on the device A. Such an operation can be written in the 

Kraus form 

where the Kraus operators M$_{i}$ satisfy 

<img src='content_image/372506.jpg'>

<img src='content_image/372507.jpg'>

Kraus operators can be chosen in various different ways[21]. However, a representation that has the smallest number n of terms will be preferred in this article. In this case, the set of Kraus operators { M$_{1}$, . . ., M$_{n}$ } are linearly independent. 

Now, let U$_{AB}$ be the isometry for a realization of the operation E . This isometry can be 

written as 

<img src='content_image/372508.jpg'>

for some operators L$_{i}$ acting on the Hilbert space of the bath B. This can be shown by completing the set of operators { M$_{1}$, M$_{2}$, . . ., M$_{n}$ } to a basis for the operator space by 

