We study the formulation of the uncertainty principle in quantum mechanics in terms of entropic inequalities, extending results recently derived by BialynickiBirula [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 Rényi 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 α and β in the plane (α, β). Our results explain and extend a recent study by Luis [3], where states with quantum fluctuations below the Gaussian case are discussed at the single point (2, 2).

We propose an alternative measure of quantum uncertainty for pairs of arbitrary observables in the 2-dimensional case, in terms of collision entropies. We derive the optimal lower bound for this entropic uncertainty relation, which results in an analytic function of the overlap of the corresponding eigenbases. Besides, we obtain the minimum uncertainty states. We compare our relation with other formulations of the uncertainty principle.Comment: The manuscript has been accepted for publication as a Regular Article in Physical Review