Holographic Software for Quantum Networks
  We introduce a pictorial approach to quantum information, called holographic software. Our software captures both algebraic and topological aspects of quantum networks. It yields a bi-directional dictionary to translate between a topological approach and an algebraic approach. Using our software, we give a topological simulation for quantum networks. The string Fourier transform (SFT) is our basic tool to transform product states into states with maximal entanglement entropy. We obtain a pictorial interpretation of Fourier transformation, of measurements, and of local transformations, including the $n$-qudit Pauli matrices and their representation by Jordan-Wigner transformations.   We use our software to discover interesting new protocols for multipartite communication. In summary, we build a bridge linking the theory of planar para algebras with quantum information. 
Arthur Jaffe, Zhengwei Liu, and Alex Wozniakowski
quant-ph hep-th math-ph math.MP math.OA math.QA