Beyond the standard entropic inequalities: stronger scalar separability criteria and their applications

Recently it was shown that if a given state fulfils the reduction criterion it must also satisfy the known entropic inequalities. Now the questions arises whether on the assumption that stronger criteria based on positive but not completely positive maps are satisfied, it is possible to derive some scalar inequalities stronger than the entropic ones. In this paper we show that under some assumptions the extended reduction criterion [H.-P. Breuer, Phys. Rev. Lett 97, 080501 (2006); W. Hall, J. Phys. A 40, 6183 (2007)] leads to some entropic--like inequalities which are much stronger than their entropic counterparts. The comparison of the derived inequalities with other separability criteria shows that such approach might lead to strong scalar criteria detecting both distillable and bound entanglement. In particular, in the case of SO(3)-invariant states it is shown that the present inequalities detect entanglement in regions in which entanglement witnesses based on extended reduction map fail. It should be also emphasized that in the case of $2\otimes N$ states the derived inequalities detect entanglement efficiently, while the extended reduction maps are useless when acting on the qubit subsystem. Moreover, there is a natural way to construct a many-copy entanglement witnesses based on the derived inequalities so, in principle, there is a possibility of experimental realization. Some open problems and possibilities for further studies are outlined.