Science China Mathematics

, Volume 61, Issue 4, pp 593–626 | Cite as

Holographic software for quantum networks

  • Arthur Jaffe
  • Zhengwei Liu
  • Alex Wozniakowski
Open Access


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.


quantum information picture language string Fourier transform Clifford gate multipartite entanglement quantum protocol 


81P45 94A15 90B18 94A05 94A17 



This work was supported by the Templeton Religion Trust (Grant Nos. TRT0080 and TRT0159). The authors are grateful for hospitality at the Research Institute for Mathematics (FIM) of the ETH-Zurich, at the Max Planck Institute for Mathematics in Bonn, at the Hausdorff Institute for Mathematics in Bonn, at the Isaac Newton Mathematical Institute in Cambridge, UK, and at the Mathematical Research Institute Oberwolfach, where they did part of this work. The authors thank Erwin Engeler, Klaus Hepp, Daniel Loss, Renato Renner, and Matthias Troyer for helpful discussions. The authors are also grateful to Bob Coecke for sharing a copy of [15], before its publication.


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Authors and Affiliations

  1. 1.Departments of Mathematics and PhysicsHarvard UniversityCambridgeUSA

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