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We study experimentally accessible lower bounds on entanglement measures based on entropicuncertainty relations. Experimentally quantifying entanglement is highly desired for applications ofquantum simulation experiments to fundamental questions, e.g. in quantum statistical mechanicsand condensed matter physics. At the same time it poses a significant challenge as the evaluationof entanglement measures typically requires the full reconstruction of the quantum state, which isextremely costly in terms of measurement statistics. We derive an improved entanglement boundfor bipartite systems, which requires measuring joint probability distributions in only two differentmeasurement settings per subsystem, and demonstrate its power by applying it to currently oper-ational experimental setups for quantum simulation with cold atoms. Examining the tightness ofthe derived entanglement bound, we find that the set of pure states for which our relation is tightis strongly restricted. We show that for measurements in mutually unbiased bases the only purestates that saturate the bound are maximally entangled states on a subspace of the bipartite Hilbertspace (this includes product states). We further show that our relation can also be employed forentanglement detection using generalized measurements, i.e. when not all measurement outcomescan be resolved individually by the detector. In addition, the impact of local conserved quantitieson the detectable entanglement is discussed.