Entanglement Detection by Local Orthogonal Observables

We propose a family of entanglement witnesses and corresponding positive maps that are not completely positive based on local orthogonal observables. As applications the entanglement witness of the $3\times 3$ bound entangled state [P. Horodecki, Phys. Lett. A {\bf 232}, 333 (1997)] is explicitly constructed and a family of $d$-dimensional bound entangled states is designed so that the entanglement can be detected by permuting local orthogonal observables. Further the proposed physically not implementable positive maps can be physically realized by measuring a Hermitian correlation matrix of local orthogonal observables.