Some extensions of the uncertainty principle

We study the formulation of the uncertainty principle in quantum mechanics in terms of entropic inequalities, extending results recently derived by Bialynicki-Birula [I. Bialynicki-Birula, Formulation of the uncertainty relations in terms of the Rényi entropies, Physical Review A 74 (5) (2006) 052101] and Zozor et al. [S. Zozor, C. Vignat, On classes of non-Gaussian asymptotic minimizers in entropic uncertainty principles, Physica A 375 (2) (2007) 499–517]. 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 [A. Luis, Quantum properties of exponential states, Physical Review A 75 (2007) 052115], where states with quantum fluctuations below the Gaussian case are discussed at the single point (2,2).