Tensor rank and entanglement of pure quantum states

Abstract

The rank of a tensor is analyzed in context of quantum entanglement. A pure quantum state $\mathbf{v}$ 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 characterization of orbits of quantum states generated by the group of special linear transformations. A quantum state $\mathbf{v}$ is called {\sl entangled}, if it {\sl cannot} be written in the product form, $\mathbf{v}\ne {\mathbf{v}}_1\otimes {\mathbf{v}}_2\otimes \cdots \otimes {\mathbf{v}}_d$, 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..

63 pages

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• Entanglement in the symmetric subspace: mapping multipartite to bipartite states
• Absolutely maximally entangled pure states of multipartite quantum systems
• Waring decompositions of the product of two quadrics: the small rank cases