Quantum Lunch:Tensor surgery and tensor rank

Speaker: Prof. Matthias Christandl

Title:Tensor surgery and tensor rank

Abstract: We introduce tensor surgery and apply it to tensors defined by graphs and hypergraphs. By splitting vertices, creating virtual edges and by inserting new ones, we obtain nontrivial upper bounds on tensor rank, border rank and asymptotic rank of the resulting tensors. We illustrate our method with a number of examples. Tensor surgery on the triangle graph, which corresponds to the matrix multiplication tensor, leads to nontrivial rank bounds for all odd cycle graphs, which correspond to the tensors of iterated matrix multiplication. In the asymptotic setting we obtain upper bounds in terms of the matrix multiplication exponent ω and the rectangular matrix multiplication parameter α that are optimal if ω equals two.

We also give examples that illustrate that tensor surgery on general graphs might involve the absorption of virtual hyperedges and provide an example of tensor surgery on a hypergraph. In the context of quantum information theory, our results may be interpreted as upper bounds on the rate of Greenberger-Horne-Zeilinger (GHZ) states needed in order to create a graph of Einstein-Podolsky-Rosen states, or, more generally, GHZ states shared among subsets of the vertices.