Before revision:  Here we show that any conditional distribution P can however be implemented if the master equation dynamics has access to additional "hidden" states , not in X . We investigate how the minimal number of such hidden states needed to implement some P in a thermodynamically reversible manner depends on P. We provide exact results in the special case of conditional distributions that represent single-valued functions.
After revision:  Here we show that any conditional distribution P can however be implemented if the master equation dynamics has access to additional "hidden" states , not in X . We investigate how the minimal number of such hidden states needed to do so. We calculate this cost exactly for the special case of conditional distributions that represent single-valued functions.