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Generalised Compositional Theories and Diagrammatic Reasoning

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Quantum Theory: Informational Foundations and Foils

Part of the book series: Fundamental Theories of Physics ((FTPH,volume 181))

Abstract

This chapter provides an introduction to the use of diagrammatic language, or perhaps more accurately, diagrammatic calculus, in quantum information and quantum foundations. We illustrate the use of diagrammatic calculus in one particular case, namely the study of complementarity and non-locality, two fundamental concepts of quantum theory whose relationship we explore in later part of this chapter.

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Notes

  1. 1.

    We jest; reading Mac Lane’s book is eventually unavoidable, however the paper [18] is an easy introduction to the subject of monoidal categories.

  2. 2.

    The term “basic” simply means a process whose internal structure is of no interest. Typically we construct diagrams from some given set of basic processes.

  3. 3.

    This is actually true even for non-symmetric monoidal categories; see [7].

  4. 4.

    Since we identify sets of the same cardinality, we can equivalently say that the systems of \(\mathbf {FRel}\) are just the natural numbers.

  5. 5.

    To show completeness for a rewrite theory it is typically necessary, but rarely sufficient, to check that the rewrite rules are confluent; that is, whenever two rewrites simultaneously apply to a given diagram, then the choice between then (eventually) does not matter. Since this property must hold for every diagram and every pair of rewrites, even a simple rewrite system can produce an extremely large number of cases, necessitating a computer-assisted proof. For example see the work of Lafont on Boolean circuits [20].

  6. 6.

    In other words, rank 1 projectors.

  7. 7.

    For the experts in category theory, this additional structure can be summed up by saying we operate in a dagger-compact category, rather than just a symmetric monoidal category.

  8. 8.

    Indeed Eqs. (6) and (11) are equivalent in any theory wherever at least one of the observable structures has enough classical points.

  9. 9.

    For a formal statement and proof of this theorem, in terms of factorisation systems see [36].

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Authors and Affiliations

  1. Department of Computer Science, University of Oxford, Wolfson Building, Parks Road, Oxford, OX1 3QD, UK

    Bob Coecke & Aleks Kissinger

  2. Department of Computer and Information Sciences, University of Strathclyde, Livingston Tower, 26 Richmond Street, Glasgow, G1 1XH, UK

    Ross Duncan

  3. School of Mathematics and System Sciences, Beihang University, XueYuan Road No.37, HaiDian District, Beijing, China

    Quanlong Wang

Authors
  1. Bob Coecke
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  2. Ross Duncan
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  4. Quanlong Wang
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Correspondence to Bob Coecke .

Editor information

Editors and Affiliations

  1. Department of Computer Science, The University of Hong Kong, Hong Kong, China

    Giulio Chiribella

  2. Perimeter Inst Theoretical Physics , Waterloo, Ontario, Canada

    Robert W. Spekkens

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Coecke, B., Duncan, R., Kissinger, A., Wang, Q. (2016). Generalised Compositional Theories and Diagrammatic Reasoning. In: Chiribella, G., Spekkens, R. (eds) Quantum Theory: Informational Foundations and Foils. Fundamental Theories of Physics, vol 181. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-7303-4_10

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