Multipartite bound entanglement and multi-setting Bell inequalities

Dür [Phys. Rev. Lett. 87, 230402 (2001)] constructed N-qubit bound entangled states which violate a Bell inequality for N ≥ 8, and his result was recently improved by showing that there exists an N-qubit bound entangled state violating the Bell inequality if and only if N ≥ 6 [Phys. Rev. A 79, 032309 (2009)]. On the other hand, it has been also shown that the states which Dür considered violate Bell inequalities different from the inequality for N ≥ 6. In this paper, by employing different forms of Bell inequalities, in particular, a specific form of Bell inequalities with M settings of the measuring apparatus for sufficiently large M , we prove that there exists an N-qubit bound entangled state violating the M -setting Bell inequality if and only if N ≥ 4.