$\rho_{AB}$, find the eigenvalues of any of its partial transpositions. A negative eigenvalue immediately implies that the state is entangled. Examples of states for which the partial transposition has negative eigenvalues include the singlet state.

The partial transposition criterion allows to detect in a straightforward manner all entangled states that are NPT states. This is a huge class of states. However, it turns out that there exist PPT states which are not separable, as pointed out in Ref. [@Pawelbound] (see also [@Horodeckibound]). Moreover, the set of PPT entangled states is not a set of measure zero [@koto-volume-re]. It is, therefore, important to have further independent criteria of entanglement detection which permits to detect entangled PPT states. It is worth mentioning here that PPT states which are entangled, form the only known examples of the “bound entangled states” (see Refs. [@Horodeckibound; @biyog] for details). Bound entangled states of bipartite quantum states are the states that cannot be *distilled* i.e., converted to singlet states under LOCC [@bennett_dist; @bennett_eof2], with other entangled states being distillable. We will talk about distillation of quantum states later in this chapter in a bit more detail. Although as yet not found, it is conjectured that there also 