Uncertainty Relations for General Canonically Conjugate Observables in Terms of Unified Entropies

Abstract

We study uncertainty relations for a general class of canonically conjugate observables. It is known that such variables can be approached within a limiting procedure of the Pegg–Barnett type. We show that uncertainty relations for conjugate observables in terms of generalized entropies can be obtained on the base of genuine finite-dimensional consideration. Due to the Riesz theorem, there exists an inequality between norm-like functionals of two probability distributions in finite dimensions. Using a limiting procedure of the Pegg–Barnett type, we take the infinite-dimensional limit right for this inequality. Hence, uncertainty relations are derived in terms of generalized entropies. In particular, the case of measurements with a finite precision is addressed. It takes into account that in any experiment an accuracy of performed measurements is always limited.