Some informational quantifiers' inequalities for the quantum oscillator compatible with a minimal observable length

PUERTAS CENTENO, D.; PORTESI, M.

Boulder, Colorado

Conferencia; Quantum Information Processing (QIP 2019); 2019

University of Colorado at Boulder

The quantum mechanics' uncertainty principle can be expressed under many di erent forms, ranging from the well-known Heisenberg-Robertson inequalities for the product of variances of noncommuting operators, to entropic uncertainty relations in terms of Renyi or other entropy functionals, to inequalities involving Fisher information or generalized moments. Actually, these uncertainty relations are related with complexity measures. Interestingly enough, some of these relations, and also have been extended in the presence of quantum memory.Here we examine generalizations of the position-momentum uncertainty relation, that take into account the possibility of observation of a minimal length as proposed in quantum gravity. This appears as a deformation in the commutator between the operators for position and momentum, such as [X, P ] = i (1 + beta P^2) with a positive parameter beta. For the 1D harmonic oscillator Hamiltonian, the solutions of the Schrodinger equation in P-space are found. Informational quantifiers as Renyi entropy, Fisher information and the corresponding complexity measures are computed for momentum and position spaces, and a set of inequalities is obtained. The state that saturates the Fisher-Renyi inequalities is obtained for given values of the deformation and entropic indices, thus providing a connection between those parameters.