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

Abstract

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 form U_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 inequalities.