Some extensions of the uncertainty principle

Abstract

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).

Position-momentum uncertainty relations based on moments of arbitrary order

“…Finally, let us also mention that Zozor et al [20] showed that for β > α * no uncertainty principle exists, in the sense that the product of Rényi entropy powers is just trivially non-negative. But below the conjugation curve, it is not known yet neither the sharpest bound, nor the states that saturate the uncertainty relation.…”

Effect of fluctuation measures on the uncertainty relations between two observables: Different measures lead to opposite conclusions

“…In this work we are not interested in the precise lower bounds of q , q , or U q [9]. Instead we are worried by contradictions between the conclusions derived from different choices of q.…”

“…These measures may enter in uncertainty relations for two observables A,B via nontrivial lower bounds on different combinations [4][5][6][7][8][9], such as the sum of Tsallis entropies…”

“…For example: (i) variance is not always a well behaved estimator of fluctuations beyond Gaussian statistics [1], (ii) in finite-dimensional systems there are no nontrivial lower bounds for the product of variances, since the variance of an observable can vanish while the variance of any other is bounded from above [2], (iii) for periodic variables such as angle and phase variance is ambiguous and rather useless by strongly depending on the angle or phase window [3], and (iv) there are observables not easily represented by Hermitian operators [3]. This has prompted the introduction of alternative measures of fluctuations and uncertainty relations [2][3][4][5][6][7][8][9].…”

Bounds of the Pinsker and Fannes Types on the Tsallis Relative Entropy

“…Generalized entropies have also found use as alternate measures of an informational content. For instance, the entropic uncertainty principle has been expressed in terms of both the Tsallis [27,35] and Rényi entropies [38,48]. Studies of generalized entropies allow to treat properties of the standard entropy in more general setting.…”