Coherent Control and Distinguishability of Quantum Channels via PBS-Diagrams

Authors Cyril Branciard , Alexandre Clément , Mehdi Mhalla , Simon Perdrix

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Author Details

Cyril Branciard
  • Université Grenoble Alpes, CNRS, Grenoble INP, Institut Néel, F-38000 Grenoble, France
Alexandre Clément
  • Université de Lorraine, CNRS, Inria, LORIA, F-54000 Nancy, France
Mehdi Mhalla
  • Université Grenoble Alpes, CNRS, Grenoble INP, LIG, F-38000 Grenoble, France
Simon Perdrix
  • Université de Lorraine, CNRS, Inria, LORIA, F-54000 Nancy, France

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Cyril Branciard, Alexandre Clément, Mehdi Mhalla, and Simon Perdrix. Coherent Control and Distinguishability of Quantum Channels via PBS-Diagrams. In 46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 202, pp. 22:1-22:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)


Even though coherent control of quantum operations appears to be achievable in practice, it is still not yet well understood. Among theoretical challenges, standard completely positive trace preserving (CPTP) maps are known not to be appropriate to represent coherently controlled quantum channels. We introduce here a graphical language for coherent control of general quantum channels inspired by practical quantum optical setups involving polarising beam splitters (PBS). We consider different situations of coherent control and disambiguate CPTP maps by considering purified channels, an extension of Stinespring’s dilation. First, we show that in classical control settings, the observational equivalence classes of purified channels correspond to the standard definition of quantum channels (CPTP maps). Then, we propose a refinement of this equivalence class generalising the "half quantum switch" situation, where one is allowed to coherently control which quantum channel is applied; in this case, quantum channel implementations can be distinguished using a so-called transformation matrix. A further refinement characterising observational equivalence with general extended PBS-diagrams as contexts is also obtained. Finally, we propose a refinement that could be used for more general coherent control settings.

Subject Classification

ACM Subject Classification
  • Theory of computation → Quantum computation theory
  • Theory of computation → Axiomatic semantics
  • Theory of computation → Categorical semantics
  • Hardware → Quantum computation
  • Hardware → Quantum communication and cryptography
  • Quantum Computing
  • Diagrammatic Language
  • Quantum Control
  • Polarising Beam Splitter
  • Categorical Quantum Mechanics
  • Quantum Switch


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