[18] J. D. Norton, preprint available from http://www.pitt.edu/˜jdnorton/papers/Waiting.pdf . [19] L. Szilard, Z. Phys. 53 , 840 (1929). [20] P. Talkner, E. Lutz and P. H¨ anggi, Phys. Rev. E 75 , 050102 (2007). [21] M. A. Nielsen and I. L. Chuang, Quantum Computation and Information (Cambridge Uni- versity Press, Cambridge, 2000). [22] R. Bhatia, Positive Definite Matrices (Princeton University Press, Princeton and Oxford, 2007). [23] For classical devices, the analogous feature of the transformation on the combined system is the fact that this is a canonical map, which preserves phase-space volumes by Liouville’s theorem. LEP can be deduced by following a similar line of reasoning. [24] In this work, it is required that the bath is initially in a canonical equilibrium state. Therefore all pure states of B are possible as initial states and the map U$_{AB}$ should send all of these to a state where A is in | ψ$_{0}$ 〉 state. [25] It should be mentioned that the dimension d of the Hilbert space of A can be smaller that r . In such cases, the reader may imagine that the initial state ρ$_{A}$ of the device is always chosen from a d -dimensional subspace of the Hilbert space of A. With this interpretation, the HTO is an operator defined only on this subspace. 33