A tensor is a multiarray with \(d\ge 3\) indices, which is a vector in the tensor product of \(d\)-vector spaces. The rank of a tensor is a minimal number of summands in a decomposition to a sum of rank-one tensors. In this talk we discuss the notions of the generic rank, maximal rank, border rank, symmetric rank and nuclear rank of tensors. We review some known results, open problems, and numerical methods to compute different ranks.

The rank of a tensor is a simple measure of quantum entanglement. A pure quantum state \(\mathbf{v}\) of a composite system consisting of \(d\) subsystems with \(n\) levels each. It 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 \(n\). A quantum state \(\mathbf{v}\) is called entangled if its not a rank-one tensor: \({\mathbf{v}} \ne \mathbf{v}_1 \otimes \mathbf{v}_2 \otimes \cdots \otimes \mathbf{v}_d\), which 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.

This is joint work with Wojciech Bruzda and Karol Życzkowski (Jagiellonian University, Krakow).


[1]    Wojciech Bruzda, Shmuel Friedland, Karol Życzkowski. Tensor rank and entanglement of pure quantum states. ArXiv:1912.06854, version 4, 63 pages, 2022.

University of Illinois at Chicago