O-reduction criterion for the separability of quantum states

We propose a family of entanglement witnesses. These witnesses are constructed by using local orthogonal observables, and therefore can be easily measured by means of local measurements and classical communications. From the Jamiokowski isomorphism we obtain the corresponding positive maps that are not completely positive. These maps lead to a new separability criterion——the O-reduction criterion, which includes the reduction criterion and the realignment criterion as special cases. In addition, we find that the O-reduction criterion can be physically realized by measuring a Hermitian correlation matrix of local orthogonal observables. As applications, we construct an explicit entanglement witness for the first bound entangled state found by Horodecki in 1997,and as well introduce a family of dd bound entangled states, whose entanglement can be detected by permuting local orthogonal observables.