Entropic uncertainty relations for mutually unbiased equiangular tight frames assigned to quantum <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>t</mml:mi> </mml:math> -designs

Building on recent significant interest in entropic uncertainty relations for quantum measurements, this work explores the entropic uncertainty relation for the mutually unbiased equiangular tight frame assigned to quantum $t$-designs. Combining the mathematical structures of quantum measurements with those of quantum designs, uncertainty relations are derived in terms of Tsallis and R\'enyi $\ensuremath{\alpha}$ entropies, and rigorous theoretical analysis demonstrates the tightness of the derived entropic lower bounds. Furthermore, a specific relation between quantum design and the index of coincidence is inferred from the information diagrams, which is crucial for deriving the novel entropic uncertainty relation. The results reveal that the entropic lower bounds can be optimized by adjusting the parameters of the quantum $t$-design, broadening the theoretical framework of the entropic uncertainty relation and providing a practical tool for quantum information tasks.