Thermodynamic Uncertainty Relation for Arbitrary Initial States

The thermodynamic uncertainty relation (TUR) describes a trade-off relation between nonequilibrium currents and entropy production and serves as a fundamental principle of nonequilibrium thermodynamics. However, currently known TURs presuppose either specific initial states or an infinite-time average, which severely limits the range of applicability. Here we derive a finite-time TUR valid for arbitrary initial states from the Cram\'er-Rao inequality. We find that the variance of an accumulated current is bounded by the instantaneous current at the final time, which suggests that ``the boundary is constrained by the bulk". We apply our results to feedback-controlled processes and successfully explain a recent experiment which reports a violation of a modified TUR with feedback control. We also derive a TUR that is linear in the total entropy production and valid for discrete-time Markov chains with non-steady initial states. The obtained bound exponentially improves the existing bounds in a discrete-time regime.