n = 2 Kraus operators 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, 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. 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 23 V. CONCLUSIONS B$_{t}$ = max ρ A ( S ( ρ$_{A}$ ) − S ( E$_{t}$ ( ρ$_{A}$ ))) , (50) Q = 2 ∑ i =1 q$_{i}$M$_{i}$ ( t ) $^{†}$M$_{i}$ ( t ) , (51) βQ ≈ t $^{2}$( 1$_{A}$ + ln(1 /t $^{2}$) X $^{†}$X ) (52) M$_{1}$ ( t ) = tX , (47) M$_{2}$ ( t ) = √ 1 − t $^{2}$X $^{†}$X , (48) E$_{t}$ ( ρ$_{A}$ ) = 2 ∑ i =1 M$_{i}$ ( t ) $^{†}$ρ$_{A}$ M$_{i}$ ( t ) , (49)