Entropic uncertainty relations for qubits

BOSYK, GUSTAVO MARTÍN; PORTESI, MARIELA; ZOZOR, STEEVE

Paraty

Workshop; IV Quantum Information, School and Workshop; 2013

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 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, which is the commonly method used in related literature. In addition, we present an analytical expression for the tight bound inside the square [0 ; 1]^2 in the 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 and suboptimal bound for any couple of indices. In all cases, we provide the minimizing states.