Quasirandom Quantum Channels

Authors Tom Bannink, Jop Briët, Farrokh Labib, Hans Maassen

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Author Details

Tom Bannink
  • CWI, 1098 XG Amsterdam, Netherlands
  • QuSoft, Science Park 123, 1098 XG Amsterdam, Netherlands
Jop Briët
  • CWI, 1098 XG Amsterdam, Netherlands
  • QuSoft, Science Park 123, 1098 XG Amsterdam, Netherlands
Farrokh Labib
  • CWI, 1098 XG Amsterdam, Netherlands
  • QuSoft, Science Park 123, 1098 XG Amsterdam, Netherlands
Hans Maassen
  • QuSoft, Science Park 123, 1098 XG Amsterdam, Netherlands
  • Korteweg-de Vries Institute for Mathematics, Radboud University, Nijmegen, Netherlands


We would like to thank Māris Ozols, Michael Walter and Freek Witteveen for fruitful discussions.

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Tom Bannink, Jop Briët, Farrokh Labib, and Hans Maassen. Quasirandom Quantum Channels. In 15th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 158, pp. 5:1-5:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)


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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Quantum information theory
  • Quantum channels
  • quantum expanders
  • quasirandomness


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