  Any quantum operation applied on a physical system is performed as a unitary
transformation on a larger extended system. If the extension used is a heat
bath in thermal equilibrium, the concomitant change in the state of the bath
necessarily implies a heat exchange with it. The dependence of the average heat
transferred to the bath on the initial state of the system can then be found
from the expectation value of a hermitian operator, which is named as the heat
transfer operator (HTO). The purpose of this article is the investigation of
the relation between the HTOs and the associated quantum operations. Since, any
given quantum operation on a system can be realized by different baths and
unitaries, many different HTOs are possible for each quantum operation. On the
other hand, there are also strong restrictions on the HTOs which arise from the
unitarity of the transformations. The most important of these is the Landauer
erasure principle. This article is concerned with the question of finding a
complete set of restrictions on the HTOs that are associated with a given
quantum operation. An answer to this question has been found only for a subset
of quantum operations. For erasure operations, these characterizations are
equivalent to the generalized Landauer erasure principle. For the case of
generic quantum operations however, it appears that the HTOs obey further
restrictions which cannot be obtained from the entropic restrictions of the
generalized Landauer erasure principle.
