## 5 Quantum Heating and Quantum Cooling Due to Quantum Environment--Kurizki's Model
Hence the following results hold:  
\[\frac{d}{dt}(\langle H_{S}+H_{B}\rangle)\rfloor_{\tau+\Delta t} > 0 \tag{16}\] \[\frac{d}{dt}\langle H_{SB}\rangle\rfloor_{\tau+\Delta t} < 0\]  
These relations are equivalent to a quantum heating of the \"system + bath\" part [30], immediately after the disturbance. The inverse effect was shown to happen a little later, i.e., one predicts a quantum cooling, which should be of interest for various experimental investigations, e.g., those on quantum gas condensates. In connection with our INS experimental context, it should be observed that the neutron-H scattering process does not represent a QND measurement. Therefore, it should be mentioned that the above results have been extended to the general case of any arbitrary abrupt disturbance [31]. This special case is then applicable to the INS experimental context, as the neutron-H collision does represent such an abrupt disturbance. For the INS examples of Section 4, the neutron-H collision corresponds to the abrupt measurement process (recall that the neutron corresponds to \(D\)), in which the system-bath (that is, the H\({}_{2}\)-nanotube)interaction causes a quantum heating (which corresponds to an anomalously increased H\({}_{2}\) translational energy). In the latter process, the quantum environment (i.e., the C-nanotube fragment adjacent to H\({}_{2}\)) plays a significant role. Thermodynamic Meaning of Negative Conditional Entropy, Generalized Landauer's Principle, and Interpretation of the INS Effect  
In quantum-theoretical as well as information-theoretical discussions of Quantumness of Correlations, the concept of conditional entropy is of central importance; see, e.g., [5,14]. Consider two physical (resp. information-theoretical) systems \(A\) and \(B\), and let \(S(A)\) and \(S(B)\) be their well-known physical [33] (resp. Shannon [34]) entropies; cf. [5]. Furthermore, the combined (or complete) system \(AB\) should have entropy \(S(AB)\). [PERSON] rule of classical probability theory introduces then the conditional entropy  
\[S(A|B)=S(AB)-S(B) \tag{17}\]  
which is a measure of our ignorance of \(A\), under the condition that we have knowledge of \(B\). In classical theory, for the conditional entropy holds necessarily \(S(A|B)\geq 0\). But in quantum theory, one easily shows that this is not always true [12]. For long time, this fact represented an interpretational problem. In the frame of quantum information theory, however, the negativity of conditional entropy was recently given an information-theoretical meaning with the aid of the quantum procedure known as quantum state merging; see, e.g., the review article [14]. But the meaning of it in the context of physical and/or thermodynamical processes remained unclear. The situation changed with the original work of [PERSON] et al. [13]. Analyzing the standard implications of the \"classical\" [PERSON]'s principle [35], the authors arrived to the conclusion that these implications no longer remain valid if quantum information is present; cf. also [36,37]. In the quantum case, the main finding is as follows: The work cost of information erasure is now determined by the quantum entropy of the system conditioned on the quantum information (of the system) being available to the observer (or to an observing [PERSON]'s demon)--the more an observer knows about the system, the less are the energy costs of the erasure under consideration. As one easily sees, this result emphasizes again the significance of conditional entropy of information, and thus it is fully in line with the well-known [PERSON]'s statement: \"Information is physical\" [6]. In more formal terms, the main result of reference [13] may be concisely summarized as follows. Consider a system \(A\) being correlated with a quantum memory \(Q\). Then, the work that an observer (or demon) having access to \(Q\) needs in order to carry out the erasure of the information content of \(A\), is denoted by \(W(A|Q)\).