Optimal thermodynamic uncertainty relation in Markov jump processes

We investigate the tightness and optimality of thermodynamic-uncertainty-relation (TUR)-type inequalities from two aspects, the choice of the Fisher information and the class of possible observables. We show that there exists the best choice of the Fisher information, given by the pseudo entropy production, and all other TUR-type inequalities in a certain class can be reproduced by this tightest inequality. We also demonstrate that if we observe not only generalized currents but generalized empirical measures, the TUR-type inequality becomes optimal in the sense that it achieves its equality in general nonequilibrium stationary systems. Combining these results, we can draw a hierarchical structure of TUR-type inequalities.