Schmidt-number witnesses and bound entanglement

Abstract

The Schmidt number of a mixed state characterizes the minimum Schmidt rank 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 for their optimization. Finally, we present strong evidence that all bound entangled states with positive partial transpose in ${\mathcal{C}}^{3}\ensuremath{\bigotimes}{\mathcal{C}}^{3}$ have Schmidt number 2.