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Improving thermodynamic bounds using correlations

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. The resulting correlation-TUR takes into account the correlations between a current and a non-current observable, thereby tightening the TUR. We demonstrate our finding on a model of the $\text{F}_1$-ATPase molecular motor, a Markov jump model consisting of two rings and transport through a two-dimensional channel. We find that the correlation-TUR is significantly tighter than the TUR and can be close to an equality even far from equilibrium.