On PPT States in CK ⊗ CM ⊗ CN Composite Quantum Systems

Due to the importance of quantum entangled states in quantum information and computation [1, 2, 3], much effort has been done recently towards an operational characterization of separable states [4, 5, 6]. The manifestations of mixed-state entanglement can be very subtle [7]. Till now there is no general efficient criterion in judging the separability. The Bell inequalities [8], Peres PPT criterion [9], reduction criterion [10, 11], majorization [12], entanglement witnesses [13, 14], extension of Peres criterion [15], matrix realignment [16], generalized partial transposition criterion (GPT) [17], generalized reduced criterion [18], give some necessary (and also sufficient for some special cases [13]) conditions for separability. The separability criterion in [13] is both necessary and sufficient but not operational. For low rank density matrices there are also some necessary and sufficient operational criteria of separability [19, 20].