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Holographic software for quantum networks

Science China Mathematics volume 61, pages 593–626 (2018)Cite this article

An Erratum to this article was published on 23 March 2018

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Abstract

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.

Change history

  • 23 March 2018

    The article Holographic software for quantum networks, written by Arthur Jaffe, Zhengwei Liu & Alex Wozniakowski, was originally published electronically on the publisher’s internet portal (currently SpringerLink) on 14/02/2018 without open access. With the author(s)’ decision to opt for Open Choice the copyright of the article changed in March 2018 to © The Author(s) 2018 and the article is forthwith distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, duplication, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made. The original article has been corrected.

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Acknowledgements

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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  1. Alex Wozniakowski

    Present address: Current address: School of Physical and Mathematical Sciences and Complexity Institute, Nanyang Technological University, Singapore, 637723, Singapore

Authors and Affiliations

  1. Departments of Mathematics and Physics, Harvard University, Cambridge, MA, 02138, USA

    Arthur Jaffe, Zhengwei Liu & Alex Wozniakowski

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  1. Arthur Jaffe
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  2. Zhengwei Liu
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Correspondence to Arthur Jaffe.

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The original version of this article was revised due to a retrospective Open Access order.

A correction to this article is available at https://doi.org/10.1007/s11425-018-9277-3

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Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, duplication, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made.

This article is published with open access at Springerlink.com, corrected publication 03/2018

The original article has been corrected.

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Jaffe, A., Liu, Z. & Wozniakowski, A. Holographic software for quantum networks. Sci. China Math. 61, 593–626 (2018). https://doi.org/10.1007/s11425-017-9207-3

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  • DOI: https://doi.org/10.1007/s11425-017-9207-3

Keywords

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

MSC(2010)

  • 81P45
  • 94A15
  • 90B18
  • 94A05
  • 94A17