##
RANK OF A TENSOR AND QUANTUM ENTANGLEMENT

SHMUEL FRIEDLAND

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).

### References

[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

Email address: friedlan@uic.edu