Location: Núcleo de Investigación y Desarrollo Tecnológico (NIDTEC)
Time and Date: 22/11/2016 – 11:00 – 13:00
Speaker: Dr. Mathias Christandl, University of Copenhagen
Title: On the tensor rank of networks of entangled pairs: tensor surgery and the laser method
Abstract: 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 Strassen’s 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 Péter Vrana and Jeroen Zuiddam (http://arxiv.org/abs/1603.03757 and https://arxiv.org/abs/1609.07476)
Short CV of the speaker. (from http://www.math.ku.dk/english/staff)
I received my PhD from the University of Cambridge in 2006. I have since been a Thomas Nevile Research Fellow at Magdalene College Cambridge (2006-2008), a Junior professor at the University of Munich (2008-2010) and an Assistant Professor at ETH Zurich (2010-2014). In April 2014, I moved to the University of Copenhagen as a Professor. I have been awarded an ERC Starting Grant in 2013 and have been elected member of The Royal Danish Academy of Sciences and Letters in 2016. I am a co-PI of the Villum Centre of Excellence for the Mathematics of Quantum Theory (QMATH).