Effect of fluctuation measures on the uncertainty relations between two observables: Different measures lead to opposite conclusions

We show within a very simple framework that different measures of fluctuations lead to uncertainty relations resulting in contradictory conclusions. More specifically we focus on Tsallis and Rényi entropic uncertainty relations and we get that the minimum joint uncertainty states for some fluctuation measures are the maximum joint uncertainty states of other fluctuation measures, and vice versa.

Figure 1

Plot of ${\Pi }_q=R_q\left(A\right)R_q\left(B\right)$ as a function of $\theta$ for $q=0.5$ (dashed line), $q=1$ (solid line), $q=2$ (dotted line), and $q=3$ (dash-dotted line).

Figure 2

Plot of $U_q$ as a function of $\theta$ for $q=1,1.5,2$.

Figure 3

Plots of ${\Sigma }_q=S_q\left(A\right)+S_q\left(B\right)$ as a function of $\theta$ for $q=1.8,2,2.5$.

Figure 4

Plot the second derivative at $\theta =\delta /2$ of ${\Pi }_q$ (solid line), $U_q$ (dashed line), and ${\Sigma }_q$ (dotted line) as functions of $q$.

Figure 5

Plot of ${\Pi }_q=R_q\left(A\right)R_q\left(B\right)$ as a function of $\theta$ in different scales for $q=2$ (dashed line) and $q=0.5$ (solid line) for $N=100$.

Figure 6

Plot of $U_q=S_q\left(A\right)+S_q\left(B\right)+(1-q)S_q\left(A\right)S_q\left(B\right)$ as a function of $\theta$ in different scales for $q=0.5$ (dashed line) and $q=1.5$ (solid line) and $N=100$.

Figure 7

Plot of the product of Shannon entropies $S_1\left(X\right)S_1\left(Y\right)$ (dashed line) and the sum of variances ${\left(\Delta X\right)}^2+{\left(\Delta Y\right)}^2$ (solid line) for pure Gaussian states as functions of $r$ with $\Delta X=1/\Delta Y=\exp \left(r\right)$.