From my old notes taken from a series of lectures by Seth Lloyd, I think I've seen through what his simple example is saying about the volume of a phase space. The example is first given for the function f(x,p) where the momentum p is constant (as in a digital computer), and the phase space of a single particle is like it being in one partition or the other, labeled 0 and 1 (i.e. x). That's where Maxwell's demon can enter the picture, since it has to know that a molecule of gas has a constant momentum, and that it's in one partition or the other; that's how it gets to separate "hot" molecules from "cold" ones and violate the second law of thermodynamics. Anyhoo, the problem of erasure for this inital single-molecule/two-partitions phase space is that the molecule has to be in one of the partitions; all a demon can do is forget which one. Lloyd then shows how increasing the phase space to two values for p, then means erasure and (partial) reversibility are available, depending on whether p or x is the "control input". This extended phase space has a natural model as a Boolean gate, with two outputs. But it's just a model of classical measurement. Makes sense, eh?
