  From a practical perspective it is advantageous to develop methods that
verify entanglement in quantum states with as few measurements as possible. In
this paper we investigate the minimal number of mutually unbiased bases (MUBs)
needed to detect bound entanglement in bipartite $(d\times d)$-dimensional
states, i.e. entangled states that are positive under partial transposition. In
particular, we show that a class of entanglement witnesses composed of mutually
unbiased bases can detect bound entanglement if the number of measurements is
greater than $d/2+1$. This is a substantial improvement over other detection
methods, requiring significantly fewer resources than either full quantum state
tomography or measuring a complete set of $d+1$ MUBs. Our approach is based on
a partial characterisation of the (non-)decomposability of entanglement
witnesses. We show that non-decomposability is a universal property of MUBs,
which holds regardless of the choice of complementary observables, and we find
that both the number of measurements and the structure of the witness play an
important role in the detection of bound entanglement.
