One-parameter class of uncertainty relations based on entropy power

We use the concept of entropy power to derive a one-parameter class of information-theoretic uncertainty relations for pairs of conjugate observables in an infinite-dimensional Hilbert space. This class constitutes an infinite tower of higher-order statistics uncertainty relations, which allows one in principle to determine the shape of the underlying information-distribution function by measuring the relevant entropy powers. We illustrate the capability of this class by discussing two examples: superpositions of vacuum and squeezed states and the Cauchy-type heavy-tailed wave function.

Figure 1

Plot of $N_{1+t}\left(x\right)N_{1+r}\left(p\right)$ (in units of ${\hslash }^2$) for the state $|{\psi }_{\zeta }\rangle$ as a function of ${log}_{10}(1+r)$ and different values of the squeezing parameter, $\zeta$. The lower bound ${\hslash }^2/4$ is saturated for both $N_{\infty }\left(x\right)N_{1/2}\left(p\right)$ and $N_{1/2}\left(x\right)N_{\infty }\left(p\right)$. For other indices, REPURs deviate from the bound with the maximum deviation at $r=0$, which corresponds to Shannon's EP.