Rank of a tensor and quantum entanglement

The rank of a tensor is analysed in the context of quantum entanglement. A pure quantum state v of a composite system consisting of d subsystems with n levels each is viewed as a vector in the d-fold tensor product of n-dimensional Hilbert space and can be identified with a tensor with d indices, each running from 1 to n. We discuss the notions of the generic rank and the maximal rank of a tensor and review results known for the low dimensions. Another variant of this notion, called the border rank of a tensor, is shown to be relevant for the characterization of orbits of quantum states generated by the group of special linear transformations. A quantum state v is called entangled, if it cannot be written in the product form, v ? v(1) ? v(2) ? . . . ? v(d) , what implies correlations between physical subsystems. A relation between various ranks and norms of a tensor and the entanglement of the corresponding quantum state is revealed.