Entropic formulation of the uncertainty principle for the number and annihilation operators

An entropic approach to formulating uncertainty relations for the number-annihilation pair is considered. We construct some normal operator that traces the annihilation operator as well as commuting quadratures with a complete system of common eigenfunctions. Expanding the measured wave function with respect to them, one obtains a relevant probability distribution. Another distribution is naturally generated by measuring the number operator. Due to the Riesz-Thorin theorem, there exists a nontrivial inequality between corresponding functionals of the above distributions. We find the bound in this inequality and further derive uncertainty relations in terms of both the Renyi and Tsallis entropies. Entropic uncertainty relations for continuous distribution as well as relations for discretized one are presented.