Now, consider the isometry V$_{ABC}$ on ABC given by It is a simple exercise to show that this isometry gives rise to the desired convex combinations, i.e., V$_{ABC}$ realizes the operation E = ∑ $_{i}$λ$_{i}$ E$_{i}$ and the associated HTO is Q = ∑ $_{i}$λ$_{i}$ Q$_{i}$ . Note that, in here, C acts like a controller system for deciding which operation should be applied on AB. The decision is given probabilistically such that E$_{i}$ is applied with probability λ$_{i}$ . Note also that the average energy of C does not change during this process. In such a case, the overall operation and the associated HTO are given by the expected convex combination expressions. (iv) As a special case of the convexity result in (iii), the set of allowed HTOs associated with a fixed quantum operation E is also convex. This set is not bounded from above by (ii); but it is bounded from below. We are primarily interested in finding lower bounds, if possible tight ones, for this set. Before investigating the relations between HTOs and quantum operations, it is appro- priate to start with the state-dependent entropic restriction of LEP. Let S ( ρ ) = − tr ρ ln ρ denote the von Neumann entropy of the density matrix ρ . For an arbitrary quantum oper- ation, the average heat emitted to the bath is bounded from below by the drop in the von Neumann entropy of the state of the device. Theorem 1 (Generalized LEP) If Q is a possible HTO for an operation E , then for all states ρ$_{A}$ of the device A. For a given ρ$_{A}$ , the equality sign in (14) holds if and only if the relation ρ ′ AB = E ( ρ$_{A}$ ) ⊗ ρ$_{B}$ is satisfied for some (and therefore for all) realization that yields E and Q . Note that the inequality above essentially gives infinitely many bounds on the HTO, i.e., there is a separate bound on Q for each possible device state ρ$_{A}$ . The generalized III. RESTRICTIONS ON HEAT TRANSFER OPERATORS A. Entropic Restriction of LEP 10 tr ρ$_{A}$Q ≥ k$_{B}$T [ S ( ρ$_{A}$ ) − S ( E ( ρ$_{A}$ )) ] , (14) V$_{ABC}$ = n ∑ i =1 U ( i ) AB ⊗ ( | i 〉 〈 i | )$_{C}$ (13)