Uncertainty relations in stochastic processes: An information inequality approach

The thermodynamic uncertainty relation is an inequality stating that it is impossible to attain higher precision than the bound defined by entropy production. In statistical inference theory, information inequalities assert that it is infeasible for any estimator to achieve an error smaller than the prescribed bound. Inspired by the similarity between the thermodynamic uncertainty relation and the information inequalities, we apply the latter to systems described by Langevin equations and derive the bound for the fluctuation of thermodynamic quantities. When applying the Cram\'er-Rao inequality, the obtained inequality reduces to the fluctuation-response inequality. We find that the thermodynamic uncertainty relation is a particular case of the Cram\'er-Rao inequality, in which the Fisher information is the total entropy production. Using the equality condition of the Cram\'er-Rao inequality, we find that the stochastic total entropy production is the only quantity which can attain equality in the thermodynamic uncertainty relation. Furthermore, we apply the Chapman-Robbins inequality and obtain a relation for the lower bound of the ratio between the variance and the sensitivity of systems in response to arbitrary perturbations.