Complementarity relations for design-structured POVMs in terms of generalized entropies of order $α\in(0,2)$

Abstract: Information entropies give a genuine way to characterize quantitatively an incompatibility in quantum measurements. Together with the Shannon entropy, few families of parametrized entropies have found use in various questions. It is also known that a possibility to vary the parameter can often provide more restrictions on elements of probability distributions. In quantum information processing, one often deals with measurements having some special structure. Quantum designs are currently the subject of active research, whence the aim to formulate complementarity relations for related measurements occurs. Using generalized entropies of order $\alpha\in(0,2)$, we obtain uncertainty and certainty relations for POVMs assigned to a quantum design. The structure of quantum designs leads to several restrictions on generated probabilities. We show how to convert these restrictions into two-sided entropic estimates. One of the used ways is based on truncated expansions of the Taylor type. The recently found method to get two-sided entropic estimates uses polynomials with flexible coefficients. We illustrate the utility of this method with respect to both the R\'{e}nyi and Tsallis entropies. Possible applications of the derived complementarity relations are briefly discussed.