Role of Quantum Coherence in Kinetic Uncertainty Relations

The kinetic uncertainty relation (KUR) bounds the signal-to-noise ratio of stochastic currents in terms of the number of transitions per unit time, known as the dynamical activity. This bound was derived in a classical context and can be violated in the quantum regime due to coherent effects. However, the precise connection between KUR violations and quantum coherence has so far remained elusive, despite significant investigation. In this Letter, we solve this problem by deriving a modified bound that exactly pinpoints how, and when, coherence might lead to KUR violations. Our bound is sensitive to the specific kind of unraveling of the quantum master equation. It, therefore, allows one to compare quantum jumps and quantum diffusion, and understand, in each case, how quantum coherence affects fluctuations. We illustrate our result on a double quantum dot, where the electron current is monitored either by electron jump detection or with continuous diffusive charge measurement.