Uncertainty relations in terms of the Tsallis entropy

Uncertainty relations emerging from the Tsallis entropy are derived and discussed. In particular we found a positively defined function that saturates the so-called entropic inequalities for entropies characterizing the physical states under consideration.

Figure 1

Example of the dependence (for $\alpha =1.3$) of limitations of the sum of entropies on the size of cell in phase space, $dxdp/h$. Results for Renyi and Shannon entropies are practically indistinguishable (to expose both the large and small values we used the linear-logarithmic scale here, in this case for both entropies one gets straight lines). Horizontal lines indicate the corresponding bounds for limitation imposed on Tsallis entropies for which $1/\left(1-\alpha \right)<\text{inf}\left\{H^{\left(x\right)}+H^{\left(p\right)}\right\}<1/\left(\alpha -1\right)$.

Figure 2

Examples of magnitude of the bias for the Gaussian shape of probability densities, $\tilde{\rho }\left(p\right)={\left|\tilde{\psi }\left(p\right)\right|}^2$ (calculated for $\alpha =1.3$ and $k=0$). Broken line shows the ratio of the right to the left side of inequality (23) whereas the solid line shows the same for inequality (28) where we put $\delta t=\delta p/\left(s_p+\delta p\right)$.

Figure 3

Example of the dependence (for $\alpha =1.3$) of limitations of the sum of entropies on the size of cell in phase space, $\delta x\delta p/h$. Results for Renyi and Shannon entropies are practically indistinguishable. Horizontal lines indicate corresponding limits for the sum of Tsallis entropies for which $1/\left(\alpha -1\right)\le \text{inf}\left\{H_{\alpha }^{\left(p\right)}+H_{\beta }^{\left(x\right)}\right\}\le \left[1-{\left(2\alpha \right)}^{-1/2\alpha }{\left(2\beta \right)}^{1/2\beta }\right]/\left(\alpha -1\right)$.