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

TL;DR

This paper derives tight, state-independent uncertainty and certainty relations for the three Pauli observables using Re9nyi entropies of order b5a1(0;1], providing conditions for equality and novel bounds for pure states.

Contribution

It introduces new tight bounds and conditions for equality in Re9nyi-entropy uncertainty relations for Pauli observables, including pure-state bounds and a comparison with Tsallis entropy.

Findings

Bounds are tight and always reached by certain pure states.

Conditions for equality in Re9nyi-entropy uncertainty relations are established.

The width of the entropic band for pure states is compared with Tsallis entropy results.

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.