Optimal uncertainty relations for extremely coarse-grained measurements

We derive two quantum uncertainty relations for position and momentum coarse-grained measurements. Building on previous results, we first improve the lower bound for uncertainty relations using the Rényi entropy, particularly in the case of coarse-grained measurements. We then sharpen a Heisenberg-like uncertainty relation derived previously in [Europhys. Lett. 97, 38003 (2012)] that uses variances and reduces to the usual one in the case of infinite precision measurements. Our sharpened uncertainty relation is meaningful for any amount of coarse graining. That is, there is always a nontrivial uncertainty relation for coarse-grained measurement of the noncommuting observables, even in the limit of extremely large coarse graining.

Figure 1

Example of a continuous $w_{\Delta }\left(x\right)$ distribution function (blue dashed line) approximating the original distribution function $\rho \left(x\right)$ (black line) constructed from rectangle functions.

Figure 2

We plot the lower bounds $\mathcal{R}$ (green solid line), ${\mathcal{B}}_1$ (red dashed line), and ${\mathcal{B}}_{1/2}$ (black dashed-dotted line). Note that for $\alpha =1/2$ we have $L_{1/2}=\mathcal{R}$.

Figure 3

Plots of the $\mathcal{M}\left(t\right)$ function (black solid line) and the ${\mathcal{M}}^{-1}\left(t\right)$ inverse function (red dashed line).

Figure 4

Comparison between the function $K\left(u\right)$ (red solid line) and the linear function $1+2\pi eu$ (black dashed line).

Figure 5

Plot of the uncertainty relation of Eq. (49) as a function of the discrete variances ${\sigma }_{x_{\Delta }}^2$ and ${\sigma }_{p_{\delta }}^2$. The red forbidden region shows that there is always an uncertainty relation for any size coarse graining.