Beyond Landau-Pollak and entropic inequalities: Geometric bounds imposed on uncertainties sums

ZOZOR, STEEVE; BOSYK, GUSTAVO MARTÍN; PORTESI, MARIELA; OSÁN, TRISTÁN MARTÍN; LAMBERTI. PEDRO WALTER

Amboise

Workshop; 34th International Workshop on Bayesian Inference and Maximum Entropy Methods in Science and Engineering; 2014

In this paper we propose generalized inequalities to quantify the uncertainty principle. We deal with two observables with finite discrete spectra described by positive operator-valued measures (POVM) and with systems in mixed states. Denoting by $p(A; ho)$ and $p(B; ho)$ the probability vectors associated with observables $A$ and $B$ when the system is in the state $ ho$, we focus on relations of the form $U_alpha(p(A; ho)) + U_eta(p(B; ho)) ge B_{alpha,eta}(A,B)$ where $U_lambda$ is a measure of uncertainty and $B$ is a non-trivial state-independent bound for the uncertainty sum. We propose here: (i) an extension of the usual Landau--Pollak inequality for uncertainty measures of the form $U_f(p(A; ho)) = f(max_i p_i(A; ho))$ issued from well suited metrics; our generalization comes out as a consequence of the triangle inequality. The original Landau--Pollak inequality initially proved for nondegenerate observables and pure states, appears to be the most restrictive one in terms of the maximal probabilities; (ii) an entropic formulation for which the uncertainty measure is based on generalized entropies of R´enyi or Havrda--Charv´at--Tsallis type: $U_{g,alpha}(p(A; ho)) = rac{g(sum_i [p_i(A; ho)]^alpha)}{1-alpha}$. Our approach is based on Schur-concavity considerations and on previously derived Landau--Pollak type inequalities.