Complementarity relations for measurements assigned to quantum designs

Complementarity relations between various informational characteristics are of interest. In applied questions, one can deal with situations, where the sums of certain powers of probabilities are known. The basic question is how to convert the imposed restrictions into two-sided estimates on the Shannon entropy. There are two different ways to address the question. The first of them is based on truncated expansions of the Taylor type. Another method is based on the use of coefficients of the shifted Chebyshev polynomials. It is an example of Lancsoz's expansions with flexible coefficients. The presented methods are used to derive uncertainty and certainty relations for positive operator-valued measures assigned to a quantum design. Quantum designs are currently the subject of active researches due to potential use in emerging quantum technologies. It is shown that the derived estimates are useful in some questions of quantum information science.