entanglement witness. This is a result of geometric nature and invokes the Hahn–Banach theorem (see reference below). From the existence of entanglement witnesses, one can show that being positive for all positive maps Λ is a necessary and sufficient condition for the separability of ρ, where Λ maps to Furthermore, every positive map from to can be decomposed into a sum of completely positive and completely copositive maps, when and . In other words, every such map Λ can be written as where and are completely positive and T is the transposition map. This follows from the Størmer-Woronowicz theorem. Loosely speaking, the transposition map is therefore the only one that can generate negative eigenvalues in these dimensions. So if is positive, is positive for any Λ. Thus we conclude that the Peres–Horodecki criterion is also sufficient for separability when . In higher dimensions, however, there exist maps that can't be decomposed in this fashion, and the criterion is no longer sufficient. Consequently, there are entangled states which have a positive partial transpose. Such states have the interesting property that they are bound entangled, i.e. they can not be distilled for quantum communication purposes. Continuous variable systems The Peres–Horodecki criterion has been extended to continuous variable systems. Rajiah Simon formulated a particular version of the PPT criterion in terms of the second-order moments of canonical operators and showed that it is necessary and sufficient for -mode Gaussian states (see Ref. for a seemingly different but essentially equivalent approach). It was later found that Simon's condition is also necessary and sufficient for -mode Gaussian states, but no longer sufficient for -mode Gaussian states. Simon's condition can be generalized by taking into account the higher order moments of canonical operators or by using entropic measures. Symmetric systems For symmetric states of bipartite systems, the positivity of the partial transpose of the density matrix is related to the sign of certain two-body correlations. Here, symmetry means that holds, where is the flip or swap operator exchanging the two parties and . A full basis of the symmetric subspace is of the form with and Here for and must hold, where is the dimension of the two parties. It can be shown that for such states, has a positive partial transpose if and only if holds for all operators Hence, if holds for some then the state possesses non-PPT entanglement. References Karol Życzkowski and Ingemar