On entropic uncertainty relations for measurements of energy and its "complement"

Alexey E. Rastegin

Published 2018-07-30Version 1

Reformulation of Heisenberg's uncertainty principle in application to energy and time is a powerful heuristic principle. In a qualitative form, this statement plays an important role in foundations of quantum theory and statistical physics. A typical meaning of energy-time uncertainties is as follows. If some state exists for a finite interval of time, then it cannot have a completely definite value of energy. It is also well known that the case of energy and time principally differs from more familiar examples of two non-commuting observables. Since quantum theory was originating, many approaches to energy-time uncertainties have been proposed. Entropic approach to formulating the uncertainty principle is currently the subject of active researches. Using the Pegg concept of complementarity of the Hamiltonian, we obtain uncertainty relations of the "energy-time" type in terms of the R\'enyi and Tsallis entropies. Although this concept is somehow restricted in scope, derived relations can be applied to systems typically used in quantum information processing. Both the state-dependent and state-independent formulations are of interest. Some of the derived state-independent bounds are similar to the results obtained within a more general approach on the base of sandwiched relative entropies. In this regard, our relations provide an alternative viewpoint on such bounds. The developed approach also allows us to address the case of detection inefficiencies. It is worth for several reasons including possible information-processing applications.