Inseparability criteria based on bipartitions of $N$-spin systems

Abstract: We present a new set of inseparabilty inequalities to detect entanglement in $N$-spin states. These are based on negative partial transposition and involve collective spin-spin correlations of any two partitions of the entire system. They also reveal the rich texture of partial separability for different partition and can discriminate between GHZ-type and W-type entanglement, as well. We introduce a new concept of relative entanglement of two different systems and two different partitions in a spin-ensemble. These criteria can be applicable to non-symmetric states and states with odd or even $N$.