License: Creative Commons Attribution 3.0 Unported license (CC-BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.TQC.2020.5
URN: urn:nbn:de:0030-drops-120642
URL: https://drops.dagstuhl.de/opus/volltexte/2020/12064/
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Bannink, Tom ; Briƫt, Jop ; Labib, Farrokh ; Maassen, Hans

Quasirandom Quantum Channels

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LIPIcs-TQC-2020-5.pdf (0.5 MB)


Abstract

Mixing (or quasirandom) properties of the natural transition matrix associated to a graph can be quantified by its distance to the complete graph. Different mixing properties correspond to different norms to measure this distance. For dense graphs, two such properties known as spectral expansion and uniformity were shown to be equivalent in seminal 1989 work of Chung, Graham and Wilson. Recently, Conlon and Zhao extended this equivalence to the case of sparse vertex transitive graphs using the famous Grothendieck inequality. Here we generalize these results to the non-commutative, or "quantum", case, where a transition matrix becomes a quantum channel. In particular, we show that for irreducibly covariant quantum channels, expansion is equivalent to a natural analog of uniformity for graphs, generalizing the result of Conlon and Zhao. Moreover, we show that in these results, the non-commutative and commutative (resp.) Grothendieck inequalities yield the best-possible constants.

BibTeX - Entry

@InProceedings{bannink_et_al:LIPIcs:2020:12064,
  author =	{Tom Bannink and Jop Bri{\"e}t and Farrokh Labib and Hans Maassen},
  title =	{{Quasirandom Quantum Channels}},
  booktitle =	{15th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2020)},
  pages =	{5:1--5:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-146-7},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{158},
  editor =	{Steven T. Flammia},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/opus/volltexte/2020/12064},
  URN =		{urn:nbn:de:0030-drops-120642},
  doi =		{10.4230/LIPIcs.TQC.2020.5},
  annote =	{Keywords: Quantum channels, quantum expanders, quasirandomness}
}

Keywords: Quantum channels, quantum expanders, quasirandomness
Collection: 15th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2020)
Issue Date: 2020
Date of publication: 08.06.2020


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