Majorization approach to entropic uncertainty relations for coarse-grained observables

We improve the entropic uncertainty relations for position and momentum coarse-grained measurements. We derive the continuous, coarse-grained counterparts of the discrete uncertainty relations based on the concept of majorization. The entropic inequalities obtained involve two Rényi entropies of the same order, and thus go beyond the standard scenario with conjugated parameters. In a special case describing the sum of two Shannon entropies, the majorization-based bounds significantly outperform the currently known results in the regime of larger coarse graining, and might thus be useful for entanglement detection in continuous variables.

Figure 1

As a comparison, I plot the previously known lower bounds ${\mathcal{B}}_1$ (red dashed line) and $\mathcal{R}$ (green dash-dotted line), together with our majorization bounds (black solid lines), labeled by $n=2,3,4$. By $H_{\Delta }$ and $H_{\delta }$ I denote $H_1\left[q^{\Delta }\right]$ and $H_1\left[p^{\delta }\right]$, respectively. From the value $\Delta \delta /\hslash \approx 4.8231$ at which the black line $(n=4)$ intersects the red dashed line, our bounds improve the previously known results.