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n = 2 Kraus operators 

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where X is a non-hermitian operator such that { M † $_{i}$( t ) M$_{j}$ ( t ) } 2 i,j =1 are linearly independent. Note that in the limit t −→ 0, the operation E$_{t}$ approaches to the identity transformation. Therefore, the maximum drop in the von Neumann entropy of the device, 

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approaches zero in the same limit. Then, the smallest possible value of the coefficient α allowed by the entropic restrictions of LEP is α$_{min}$ = k$_{B}$TB$_{t}$ , which also tends to zero as t −→ 0. However, it is shown above that α > k$_{B}$T ln 2, independent of how close E$_{t}$ to the identity operation. This clearly shows that Theorem 4 (b) contains restrictions on HTOs that are not implied by the entropic LEP restrictions. 

In order to eliminate a possible misunderstanding that may arise from the previous ex- ample, let us stress on the fact that once we lift the restriction that Q is proportional to the identity, it is possible to find HTOs that approach to the zero operator when E$_{t}$ approaches to the identity operation. For example, for E$_{t}$ given above, 

with βq$_{2}$ = t 2 and βq$_{1}$ ≈ ln(1 /t $^{2}$) yields an HTO Q with 

which converges to zero as t −→ 0. 

## V. CONCLUSIONS 

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In this article, the concept of heat transfer operator is introduced and some of its prop- erties are discussed. The HTO is the main quantity needed for analyzing the energetics of quantum operations. Hence, the complete characterization of the HTOs associated with a 

