  We discuss how to use correlations between different physical observables to
improve recently obtained thermodynamics bounds, notably the
fluctuation-response inequality (FRI) and the thermodynamic uncertainty
relation (TUR). We show that increasing the number of measured observables will
always produce a tighter bound. This tighter bound becomes particularly useful
if one of the observables is a conserved quantity, whose expectation is
invariant under a given perturbation of the system. For the case of the TUR, we
show that this applies to any function of the state of the system. We
demonstrate our finding on a model of the F1-ATPase molecular motor, where we
obtain a bound that is more than a factor of 2 tighter than the one given by
the TUR.
