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