Some extensions of the uncertainty principle

TL;DR

This paper extends the entropic formulation of the quantum uncertainty principle to non-conjugated indices and both discrete and continuous systems, broadening the understanding of quantum uncertainties beyond Gaussian states.

Contribution

It generalizes entropic uncertainty inequalities to non-conjugated indices and explores their validity in discrete and continuous quantum systems.

Findings

Extended entropic inequalities for non-conjugated indices

Analyzed uncertainty relations in discrete and continuous spaces

Connected results to quantum fluctuation properties of non-Gaussian states

Abstract

We study the formulation of the uncertainty principle in quantum mechanics in terms of entropic inequalities, extending results recently derived by Bialynicki-Birula [1] and Zozor et al. [2]. Those inequalities can be considered as generalizations of the Heisenberg uncertainty principle, since they measure the mutual uncertainty of a wave function and its Fourier transform through their associated R\'enyi entropies with conjugated indices. We consider here the general case where the entropic indices are not conjugated, in both cases where the state space is discrete and continuous: we discuss the existence of an uncertainty inequality depending on the location of the entropic indices $\alpha$ and $\beta$ in the plane $(\alpha, \beta)$. Our results explain and extend a recent study by Luis [2], where states with quantum fluctuations below the Gaussian case are discussed at the single point $(2,2)$.