Therefore, in this limit, Δ approaches to 0. This shows that Δ can be chosen as small as possible. It can also be shown that the same can be done with a finite Z$_{B}$ value. Since Δ depends continuously on ν$_{k}$ and ν ′ $_{k}$, it is then possible to adjust these parameter sequences such that Δ has the value which is a strictly positive quantity by our initial assumption. In that case, Δ E$_{B}$ = tr ρQ and hence Q is the HTO associated with this realization. This completes the proof of the Theorem 2. □ [1] R. Landauer, IBM J. Res. Dev. 5 , 183 (1961). [2] C. H. Bennett, Int. J. Theor. Phys. 21 , 905 (1982). [3] O. Penrose, “Foundations of Statistical Mechanics”, (Pergamon Press, Oxford, 1970). [4] H.S. Leff and A.F. Rex, “Maxwell’s Demon 2: Entropy, Classical and Quantum Information, Computing”, (Institute of Physics, Bristol, 2003). [5] W. H. Zurek, in G.T. Moore and M. O. Scully, “Frontiers of Nonequilibrium Statistical Physics” (Plenum Press, New York, 1984); also reprinted in Ref. 4. [6] S. LLoyd, Phys. Rev. A 56 , 3374 (1997). [7] C. M. Bender, D. C. Brody and B. J. Meister, Proc. R. Soc. A 461 , 733 (2005). [8] K. Shizume, Phys. Rev. E 52 , 3495 (1995). [9] B. Piechocinska, Phys. Rev. A 61 , 062314 (2000). [10] K. Jacobs, arXiv:quant-ph/0512105v1 (2005). [11] CAI Qing-Yu, Chin. Phys. Lett. 21 , 1189 (2004). [12] O. J. E. Maroney, Studies in History and Philosophy of Modern Physics 36 , 355 (2005). [13] O. J. E. Maroney, Phys. Rev. E 79 , 031105 (2009). [14] S. Turgut, Phys. Rev. E 79 , 041102 (2009). [15] N. G. Anderson, Phys. Lett. A 372 , 5552 (2008). [16] T. Sagawa and M. Ueda, Phys. Rev. Lett. 102 , 250602 (2009). [17] H.P. Breuer and F. Petruccione, “The theory of open quantum systems” (Oxford University Press, Oxford, 2002). 32 Δ = k$_{B}$T ( J + S ( ρ$_{0}$ )) , (B20)