Optimal uncertainty relations in a modified Heisenberg algebrah

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 that are thought (but not proven) to imply the existence of a minimal observable length. Herewe prove this statement in a framework of sufficient physical and structural assumptions. Moreover, we present a general method that allows us to formulate optimal and state-independent variance-based uncertainty relations. In addition, instead of variances, wemake use of entropies as a measure of uncertainty and provide uncertainty relations in terms of min and Shannon entropies. We compute the corresponding entropicminimal lengths and find that theminimal length in terms ofmin entropy is exactly 1 bit.