A family of thermodynamic uncertainty relations valid for general fluctuation theorems

Thermodynamic Uncertainty Relations (TURs) are relations that establish lower bounds for the relative fluctuations of thermodynamic quantities in terms of the statistics of the associated entropy production. In this work we derive a family of TURs that explores higher order moments of the entropy production and is valid in any situation a Fluctuation Theorem holds. The resulting bound holds in both classical and quantum regimes and can always be saturated. These TURs are shown in action for a two level system weakly coupled to a bath undergoing a non time-symmetric drive, where we can use the Tasaki-Crooks fluctuation theorem. Finally, we draw a connection between our TURs and the existence of correlations between the entropy production and the thermodynamic quantity under consideration.