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Uncertainty relations and quantum entanglement are pivotal concepts inquantum theory. Beyond their fundamental significance in shaping ourunderstanding of the quantum world, they also underpin crucial applications inquantum information theory. In this article, we investigate entropicuncertainty relations and entanglement detection with an emphasis on quantummeasurements with design structures. On the one hand, we derive improvedR\'enyi entropic uncertainty relations for design-structured measurements,exploiting the property that the sum of powered (e.g., squared) probabilitiesof obtaining different measurement outcomes is now invariant under unitarytransformations of the measured system and can be easily computed. On the otherhand, the above property essentially imposes a state-independent upper bound,which is achieved at all pure states, on one's ability to predict localoutcomes when performing a set of design-structured measurements on quantumsystems. Realizing this, we also obtain criteria for detecting multi-partiteentanglement with design-structured measurements.

Entropic uncertainty relations and entanglement detection from quantum designs