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

TL;DR

This paper develops entropic uncertainty relations for energy and its complement using Rényi and Tsallis entropies, with applications to quantum information systems and considerations for detection inefficiencies.

Contribution

It introduces new entropic uncertainty relations based on Pegg’s complementarity concept, extending to Rényi and Tsallis entropies and addressing detection inefficiencies.

Findings

Derived state-dependent and state-independent bounds.

Relations applicable to quantum information processing systems.

Bounds similar to those from sandwiched relative entropies.

Abstract

Heisenberg's uncertainty principle in application to energy and time is a powerful heuristics. This statement plays the important role in foundations of quantum theory and statistical physics. If some state exists for a finite interval of time, then it cannot have a completely definite value of energy. It is 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 way to formulate 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\'{e}nyi 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 basis of sandwiched relative entropies. The developed method allows us to address the case of detection inefficiencies.