  Various theories that aim at unifying gravity with quantum mechanics suggest
modifications of the Heisenberg algebra for position and momentum. From the
perspective of quantum mechanics, such modifications lead to new uncertainty
relations which are thought (but not proven) to imply the existence of a
minimal observable length. Here we prove this statement in a framework of
sufficient physical and structural assumptions. Moreover, we present a general
method that allows to formulate optimal and state-independent variance-based
uncertainty relations. In addition, instead of variances, we make use of
entropies as a measure of uncertainty and provide uncertainty relations in
terms of min- and Shannon entropies. We compute the corresponding entropic
minimal lengths and find that the minimal length in terms of min-entropy is
exactly one bit.
