Entropic uncertainty relations from quantum designs

Abstract

In the course of the last decades entropic uncertainty relations have attracted much attention not only due to their fundamental role as manifestation of non-classicality of quantum mechanics, but also as major tools for applications of quantum information theory. Amongst the latter are protocols for the detection of quantum correlations or for the secure distribution of secret keys. In this work we show how to derive entropic uncertainty relations for sets of measurements whose effects form quantum designs. The key property of quantum designs is their indistinguishability from truly random quantum processes as long as one is concerned with moments up to some finite order. Exploiting this characteristic enables us to evaluate polynomial functions of measurement probabilities which leads to lower bounds on sums of generalized entropies. As an application we use the derived uncertainty relations to investigate the incompatibility of sets of binary observables.

arXiv comment12 pages, 4 figures

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• Entropic uncertainty relations from equiangular tight frames and their applications
• Semi-device-independent certification of independent quantum state and measurement devices
• Statistically significant tests of multiparticle quantum correlations based on randomized measurements
• Rényi formulation of uncertainty relations for POVMs assigned to a quantum design
• Estimating the Shannon entropy and (un)certainty relations for design-structured POVMs
• Entropic uncertainty relations for SIC-POVMs and MUMs
• Uncertainty relations for quantum measurements from generalized equiangular tight frames
• Entropic uncertainty relations for multiple measurements assigned with biased weights
• Uncertainty and certainty relations for quantum coherence with respect to design-structured POVMs
• General framework of quantum complementarity from a measurement-based perspective
• Complementarity relations for design-structured POVMs in terms of generalized entropies of order $\alpha \in (0,2)$
• Optimizing incompatible triple quantum measurements
• Experimental certification of ensembles of high-dimensional quantum states with independent quantum devices
• Spin minimum uncertainty states for refined uncertainty relations