Bounds on Predictive Capabilities of Driven Markov Systems

All living systems gather information about their environment but it remains an open question whether there are fundamental limits on this process. We used coupled Markov chains to quantify the predictive capabilities of a system that evolves under the influence of a stochastically changing environment. We assumed that the environmental states change according to a fixed transition matrix, while the single-step jump probabilities of the system depend on the state of the environment. Because of this coupling, the system’s states carry information about the future environmental states. Using the tools of information theory and chaotic dynamics, we prove that the predictive power of a system is bounded by the stored information, a dynamical invariant of the environment. Hence, there is a limit on how much systems can learn about a given environment regardless of the complexity of systems or the way that they are coupled to the environment. Interestingly, for certain detailed balanced environments, the stored information can be zero, indicating that no system can extract useful information about the future of these environments. Our results reveal a delicate interplay between energy expenditure and information-processing capabilities of coupled Markov systems.