Uncertainty and certainty relations for the Pauli observables in terms of the R\'{e}nyi entropies of order $\alpha\in(0;1]$

Abstract

We obtain uncertainty and certainty relations of state-independent form for the three Pauli observables with use of the R\'enyi entropies of order $\alpha\in(0;1]$. It is shown that these entropic bounds are tight in the sense that they are always reached with certain pure states. A new result is the conditions for equality in R\'enyi-entropy uncertainty relations for the Pauli observables. Upper entropic bounds in the pure-state case are also novel. Combining the presented bounds leads to a band, in which the rescaled average R\'enyi $\alpha$-entropy ranges for a pure measured state. A width of this band is compared with the Tsallis formulation derived previously.