Estimating the Shannon Entropy and (Un)certainty Relations for Design-Structured POVMs

Complementarity relations between various characterizations of a probability distribution are at the core information theory. In particular, lower and upper bounds for entropic function great importance. applied topics, we often deal with situations, where sums certain powers probabilities known. The main question is how to convert imposed restrictions into two-sided estimates on Shannon entropy. It addressed in two different ways. more intuitive them based truncated expansions Taylor type. Another method use coefficients shifted Chebyshev polynomials. We propose here family polynomials estimating entropy from below. As result, uniform sense that errors do not become too large particular points. presented used deriving uncertainty certainty positive operator-valued measures assigned quantum design. Quantum designs currently subject active researches due potential science. shown derived applicable tomography detecting steerability states.