Abstract
We consider the problem of how to construct a physical process over a finite state space X that applies some desired conditional distribution P to initial states to produce final states. This problem arises often in the thermodynamics of computation and nonequilibrium statistical physics more generally (e.g. when designing processes to implement some desired computation, feedback controller, or Maxwell demon). It was previously known that some conditional distributions cannot be implemented using any master equation that involves just the states in X. However, here we show that any conditional distribution P can in fact be implemented—if additional ‘hidden’ states not in X are available. Moreover, we show that it is always possible to implement P in a thermodynamically reversible manner. We then investigate a novel cost of the physical resources needed to implement a given distribution P: the minimal number of hidden states needed to do so. We calculate this cost exactly for the special case where P represents a single-valued function, and provide an upper bound for the general case, in terms of the non-negative rank of P. These results show that having access to one extra binary degree of freedom, thus doubling the total number of states, is sufficient to implement 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.) Our results also imply that for certain P that can be implemented without hidden states, having hidden states permits an implementation that generates less heat.
Highlights
These results show that having access to one extra binary degree of freedom, doubling the total number of states, is sufficient to implement 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.) Our results imply that for certain P that can be implemented without hidden states, having hidden states permits an implementation that generates less heat
Master equation dynamics over a discrete state space play a fundamental role in nonequilibrium statistical physics and stochastic thermodynamics [1,2,3], and are used to model a wide variety of physical systems
The entry Pij is the probability that the system is in state i at the final time given that it was in state j at the initial time
Summary
Master equation dynamics over a discrete state space play a fundamental role in nonequilibrium statistical physics and stochastic thermodynamics [1,2,3], and are used to model a wide variety of physical systems. We establish upper bounds on the number of hidden states required to implement any stochastic matrix P using a master equation, and establish bounds when we require that P be implemented in a thermodynamically reversible way. In the real world there will often be major constraints on the form of the master equation that can be considered, e.g., due to known properties of an observed system we wish to model using a master equation, or due to limitations on what kind of system we can build An example of the former is if we know that the system’s Hamiltonian can only couple degrees of freedom in certain restricted ways. The other appendices contain proofs that are not in the main text
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