Informational inequalities for the quantum oscillator with minimal length

PORTESI, M.; PUERTAS CENTENO, D.

CABA

Congreso; VIII Conference on Quantum Foundations: Quantum Logic & Quantum Structures; 2018

The quantum mechanics? uncertainty principle can be expressed under many different forms, rangingfrom the well-known Heisenberg-Robertson inequalities for the product of variances of noncommuting oper-ators, to entropic uncertainty relations in terms of Re ́nyi or other entropy functionals, to inequalities involvingFisher information or generalized moments. Actually, these uncertainty relations are related with complexitymeasures. Interestingly enough, some of these relations have been extended in the presence of quantum mem-ory. Here we examine generalizations of the position?momentum uncertainty relation, that take into accountthe possibility of observation of a minimal length as proposed in quantum gravity. This appears as a defor-mation in the commutator between the operators for position and momentum, such as [X, P ] = i (1 + βP 2)with a positive parameter β. For the 1D harmonic oscillator Hamiltonian, the solutions of the Schro ̈dingerequation in P -space are found. Informational quantifiers as Re ́nyi entropy, Fisher information and the corre-sponding complexity measures are computed in momentum and position spaces, and a set of inequalities isobtained. The state that saturates the Fisher-Re ́nyi inequalities is obtained for given values of the deformationand entropic indices, thus providing a connection between those fundamental parameters.