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2016 - 2017 (academic year)


:: Fri 10/7/16, 1:30pm in 6C-442

Matthias Christandl (U Copenhagen)
On the tensor rank of networks of entangled pairs: tensor surgery and the laser method

We prove upper bounds on the tensor rank of networks of entangled pairs. Any graph defines such a network by associating an entangled pair to each edge of the graph. We present two methods. First, we introduce a surgery-like procedure to transform a good decomposition of a well-chosen tensor into a good decomposition of a tensor of interest. We illustrate the method with surgery on the cycle graph, which corresponds to the iterated matrix multiplication tensor and obtain the first nontrivial rank results for large odd cycles and optimal asymptotic rank results for all cycles. Second, we generalize Strassenís laser method to higher- order tensors in order to show a nontrivial upper bound on the asymptotic rank for the complete graph. ""Per edge"" this improves on the best upper bound on the matrix multiplication exponent [LG14], for four or more vertices. In entanglement theory, our results amount to protocols for creating a network of entangled pairs from GHZ states by SLOCC. In communication complexity theory, our results imply new bounds on the nondeterministic quantum communication of equality games. Our work is inspired and tightly connected with the vast body of research on matrix multiplication. Based on joint work with Peter Vrana and Jeroen Zuiddam


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