Number of hidden states needed to physically implement a given conditional distribution

We consider the problem of implementing a given conditional distribution relating the states of a physical system at two separate times using a physical process with (potentially timeinhomogeneous) master equation dynamics. This problem arises implicitly in many nonequilibrium statistical physics scenarios, e.g., when designing processes to implement some desired computations, feedback-control protocols, and Maxwellian demons. However it is known that many such conditional distributions P over a state space X cannot be implemented using master equation dynamics over just the states in X. Here we show that any conditional distribution P can be implemented — if the process has access to additional “hidden” states, not in X. In particular, we show that any conditional distribution can be implemented in a thermodynamically reversible manner (achieving zero entropy production) if there are enough hidden states available. We investigate how the minimal number of such states needed to implement any P in a thermodynamically reversible manner depends on P . We provide exact results in the special case of conditional distributions that reduce to single-valued functions. For the fully general case, we provide an upper bound in terms of the nonnegative rank of P . In particular, we show that having access to one extra binary degree of freedom (doubling the number of states) is sufficient to carry out any P . Our results provide a novel type of bound on the physical resources needed to perform information processing—the size of a system’s state space.