Quantum uncertainty relation using coherence

Measurement outcomes of a quantum state can be genuinely random (unpredictable) according to the basic laws of quantum mechanics. The Heisenberg-Robertson uncertainty relation puts constraints on the accuracy of two noncommuting observables. The existing uncertainty relations adopt variance or entropic measures, which are functions of observed outcome distributions, to quantify the uncertainty. According to recent studies of quantum coherence, such uncertainty measures contain both classical (predictable) and quantum (unpredictable) components. In order to extract out the quantum effects, we define quantum uncertainty to be the coherence of the state on the measurement basis. We discover a quantum uncertainty relation using coherence between two measurement noncommuting bases. Furthermore, we analytically derive the quantum uncertainty relation for the qubit case with three widely adopted coherence measures, the relative entropy of coherence, the coherence of formation, and the $l_1$ norm of coherence.

Figure 1

Blue solid lines are used to limit the range of $A$ and $B$. Yellow area represents the feasible area. In this figure, $c=0.5$.

Figure 2

The various results of uncertainty relations using coherence under relative entropy measure. The blue line is our analytic bound and the green line is our optimal numerical bound.