content_image/361490.jpg content_image/361489.jpg
Now, consider the isometry V$_{ABC}$ on ABC given by 

<img src='content_image/361490.jpg'>

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. 

## III. RESTRICTIONS ON HEAT TRANSFER OPERATORS 

## A. Entropic Restriction of LEP 

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 

<img src='content_image/361489.jpg'>

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 

