The position-momentum uncertainty-like inequality based on moments of arbitrary order for $d$-dimensional quantum systems, which is a generalization of the celebrated Heisenberg formulation of the uncertainty principle, is improved here by use of the Rényi-entropy-based uncertainty relation. The accuracy of the resulting lower bound is physico-computationally analyzed for the two main prototypes in $d$-dimensional physics: the hydrogenic and oscillator-like systems.

We revisit generalized entropic formulations of the uncertainty principle for an arbitrary pair of quantum observables in two-dimensional Hilbert space. Rényi entropy is used as an uncertainty measure associated with the distribution probabilities corresponding to the outcomes of the observables. We derive a general expression for the tight lower bound of the sum of Rényi entropies for any couple of (positive) entropic indices (α, β). Thus, we have overcome the Hölder conjugacy constraint imposed on the entropic indices by Riesz–Thorin theorem. In addition, we present an analytical expression for the tight bound inside the square [0 , 1/2]² in the α–β plane, and a semi-analytical expression on the line β = α. It is seen that previous results are included as particular cases. Moreover we present a semi-analytical, suboptimal bound for any couple of indices. In all cases, we provide the minimizing states.

In this paper, we propose generalizations of the de Bruijn identity based on extensions of the Shannon entropy, Fisher information and their associated divergences or relative measures. The foundation of these generalizations are the Φ- entropies and divergences of the Csiszár (or Salicrú) class considered within a multidimensional context, including the one-dimensional case, and for several type of noisy channels characterized by a more general probability distribution beyond the well-known Gaussian noise. We found that the gradient and/or the Hessian of these entropies or divergences with respect to the noise parameters naturally give rise to generalized versions of the Fisher information or divergence, which are named as the Φ-Fisher information (divergence). The obtained identities can be viewed as further extensions of the classical de Bruijn identity. Analogously, it is shown that a similar relation holds between the Φ-divergence and an extended mean-square error, named Φ-mean square error, for the Gaussian channel.