Penrose Graphical Calculus for Tensor Network States

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I will outline our recent approach to a theory of quantum networks, which we base on the graphical tensor calculus of Penrose. We aim to use this approach as a means to unite quantum computation, condensed matter physics and tensor network states with a long history of existing methods and techniques. I will sketch the status of our approach and focus on methods we have developed to reason and control the states created by specific networks of contracted tensors, as well as tensor contractions capturing the key properties of algebraic invariants of operators and states.