  Thermodynamic uncertainty relations quantify how the signal-to-noise ratio of
a given observable is constrained by dissipation. Fluctuation relations
generalize the second law of thermodynamics to stochastic processes. We show
that any fluctuation relation directly implies a thermodynamic uncertainty
relation, considerably increasing their range of applicability. In particular,
we extend thermodynamic uncertainty relations to scenarios which include
measurement and feedback. Since feedback generally breaks time-reversal
invariance, the uncertainty relations involve quantities averaged over the
forward and the backward experiment defined by the associated fluctuation
relation. This implies that the signal-to-noise ratio of a given experiment can
in principle become arbitrarily large as long as the corresponding backward
experiment compensates, e.g. by being sufficiently noisy. We illustrate our
results with the Szilard engine as well as work extraction by free energy
reduction in a quantum dot.
