Statement of uncertainty principle for quantum measurements in terms of the Renyi entropies

The aim of the work is to give the explicit proofs of the Renyi-entropy uncertainty relations presented in the previous work [A. Rastegin, arXiv:0805.1777]. The relations with both the state-dependent and state-independent entropic bounds are proved. For a pair of POVM measurements the two relations are obtained. The first of them is generalization of the known results, whereas the second is quite alternative. It is shown that both these relations are meaty. The important case of POVM's with one-rank elements is extra discussed. The measurements designed for distinction between two non-orthogonal quantum states are considered as examples.