  A more concrete counterexample is that of Allahverdyan and Nieuwenhuizen (2001), where they argued it is possible to effect an erasure process with a lower cost than Landauer's principle suggests, but only in the low temperature quantum regime. A typical assumption in statistical mechanical proofs is that the interaction energy between a system and a heat bath can be treated as negligible (at least before the start and after the end of the process). Allahverdyan and Nieuwenhuizen's example exploits this by considering situations where the temperature is sufficiently low that the interaction energy can no longer be treated as negligible when compared to kT. In this situation the heat bath can no longer be treated as a canonical distribution, independently of the system and the standard proofs do not hold. It should be noted that while Zhang and Zhang's non-Hamiltonian demon is not considered actually physically possible, Allahverdyan and Nieuwenhuizen explicitly claimed that in the low temperature regime, Landauer's principle can, in fact, be broken.