Before revision:  In particular, we show that if there are no constraints on what master equation we can construct, then having access to one extra binary degree of freedom ( doubling the total number of states ) is sufficient to carry out any P with zero entropy production.
After revision:  In particular, we show that if there are no constraints on what master equation we can construct, then having access to one extra binary degree of freedom ( doubling the total number of states ) is sufficient to carry out any P with a master equation in a thermodynamically reversible way, if there are no constraints on the allowed form of the master equation. (Such constraints can greatly increase the minimal needed number of hidden states.)