INVESTIGADORES
LAMBERTI Pedro Walter
artículos
Título:
Beyond Landau?Pollak and entropic inequalities: Geometric bounds imposed on uncertainties sums
Autor/es:
STEEVE ZOZOR; G.M. BOSYK; MARIELA PORTESI; TRISTAN OSÁN; PEDRO W. LAMBERTI
Revista:
AIP CONFERENCE PROCEEDINGS
Editorial:
American Physical Society
Referencias:
Año: 2015
ISSN:
0094-243X
Resumen:
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.