On a generalized entropic uncertainty relation in the case of the qubit

Abstract

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 (alpha, beta). 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]x[0,1/2] in the alpha-beta plane, and a semi-analytical expression on the line beta=alpha. 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.