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

STEEVE ZOZOR; GUSTAVO M. BOSYK; MARIELA PORTESI; TRISTÁN M. OSÁN; PEDRO W. LAMBERTI

AIP CONFERENCE PROCEEDINGS

AMER PHYSICAL SOC

Año: 2015 vol. 1641 p. 181 - 188

0094-243X

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. On one hand, we introduce an extension of the usual Landau-Pollak inequality for uncertainty measures based on 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. One the other hand, we introduce an entropic formulation for which the uncertainty measure is based on generalized entropies of Rényi or Havrda-Charvát-Tsallis type. Our approach is based on Schur-concavity considerations and on previously derived Landau-Pollak type inequalities.