Beyond Landau-Pollak and entropic inequalities: geometric bounds imposed on uncertainties sum

ZOZOR, S.; BOSYK, G. M.; PORTESI, M.; OSÁN, T. M.; LAMBERTI, P. W.

Amboise

Workshop; AIP Conf. Proc. 1641, 181-188 (2015). (MAXENT 2014: 34th Intl. Workshop 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;ρ) and p(B;ρ) the probability vectors associated with observables A and B when the system is in the state ρ, we focus on relations of the form U_α(p(A;ρ))+U_β(p(B;ρ)) ≥ B_{α,β} (A,B) where U_λ 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 formU_f (p(A;ρ)) = f (max_i p_i(A;ρ)) 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 Rényi or Havrda-Charvát-Tsallis type: U_{g,α}(p(A;ρ)) = g(Σ_i [p_i(A;ρ)]^α )^{1−α}. Our approach is based on Schur-concavity considerations and on previously derived Landau-Pollak type inequalitie.