  The thermodynamic uncertainty relation, which establishes a universal
trade-off between nonequilibrium current fluctuations and dissipation, has been
found for various Markovian systems. However, this relation has not been
revealed for non-Markovian systems; therefore, we investigate the thermodynamic
uncertainty relation for time-delayed Langevin systems. We prove that the
fluctuation of arbitrary dynamical observables is constrained by the
Kullback--Leibler divergence between the distributions of the forward path and
its reversed counterpart. Specifically, for observables that are antisymmetric
under time reversal, the fluctuation is bounded from below by a function of a
quantity that can be identified as a generalization of the total entropy
production in Markovian systems. We also provide a lower bound for arbitrary
observables that are odd under position reversal. The term in this bound
reflects the extent to which the position symmetry has been broken in the
system and can be positive even in equilibrium. Our results hold for finite
observation times and a large class of time-delayed systems because detailed
underlying dynamics are not required for the derivation. We numerically verify
the derived uncertainty relations using two single time-delay systems and one
distributed time-delay system.
