device and the bath changes the state and hence the energy of the bath. The HTO essen- tially provides a measure of this information exchange in terms of the energy given to the bath. The nature of the operation applied on the device (e.g., whether the information in the device is erased completely or only partially, or whether randomness from the bath is introduced to the device or not) has a strong effect on the HTO. This article investigates that relation between quantum operations and the associated HTOs. This relation essen- tially follows from the unitarity of the transformation on the combined system. However, it appears that the unitarity property is unnecessarily too restrictive; all of the results in this article can also be obtained from a weaker assumption, namely that the transformation is an isometry. This assumption is made throughout the article. The quantum operation-HTO relation is a generalization of the classical LEP to the quantum domain and hence it is primarily of theoretical interest. But, it is possible that this may find some applications as well. For example, the restrictions on HTOs can be used as a way to theoretically quantify the “information erasing nature” or some other aspect of generic quantum operations. In addition to this, the current work can also be useful in the design of non-unitary operations (e.g., erasures) on technological implementations of quantum devices. If the heat emitted during the operation is desired to be minimized, it is necessary to select or adjust the bath levels and the device-bath interaction in a suitable way. The explicit construction of the erasure operations that appear in this article is appropriate for that optimization task and hence it may be a valuable guide for this purpose. The organization of the article is as follows. In Sec. II, the HTO associated with a realization of a given quantum operation is defined. The nature of the HTO and some of its fundamental properties are also discussed in this section. Sec. III is devoted to the derivation of various restrictions on the HTOs. The first one of these is the entropic bound of the generalized LEP, which can be expressed for an arbitrary quantum operation. This bound is a good starting point for the investigation of the coding-independent characterization of quantum operation-HTO relation. After that, quantum operations that completely erase the information in the device are taken up and the associated HTOs are described. These results are then used for the description of HTOs associated with extremal quantum operations. In Sec. IV, some examples are given for quantum operations that do not completely erase the information content of the device and it is shown that LEP is not sufficient for a complete characterization of HTOs associated with these operations. This indicates the usefulness of 4