The celebrated Landauer bound is the fundamental universal cost of computation: there must be dissipation of at least kBTlog2 per erasure of one bit. This result suggests that no closed Hamiltonian system can be used to encode a useful bit. Indeed, bit erasure amounts to concentrating the probability distribution in phase space, and Liouville's theorem forbids such concentration for Hamiltonian systems. However, when the system is confined to a set of measure zero, Liouville's theorem is no longer a restriction.
We show, via a simple, exactly solvable example, that a Hamiltonian and a cyclic protocol can be designed to map an energy shell to itself while concentrating the initial probability distribution on the shell. Thus, no thermodynamic cost is associated with the erasure of this Hamiltonian bit. The important restriction that the energy of the system must be known precisely is crucial for this construction. For any non-zero uncertainty, there is a fundamental minimal non-zero energy cost to the erasure, equivalent to Landauer's bound. This makes our result of only theoretical importance; however, it provides new perspectives on the peculiar properties of the microcanonical ensemble.