Schmidt-number witnesses and bound entanglement

The Schmidt number of a mixed state characterizes the minimum Schmidt lank of the pure states needed to construct it. We investigate the Schmidt number of an arbitrary mixed state by studying Schmidt-number witnesses that detect it. We present a canonical form of such witnesses and provide constructive methods far their optimization. Finally, we present strong evidence that all bound entangled states with positive partial transpose in C(3)circle timesC(3) have Schmidt number 2.