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arXiv:0908.1787quant-ph

Quantum Picturalism

Bob Coecke

Abstract

The quantum mechanical formalism doesn't support our intuition, nor does it elucidate the key concepts that govern the behaviour of the entities that are subject to the laws of quantum physics. The arrays of complex numbers are kin to the arrays of 0s and 1s of the early days of computer programming practice. In this review we present steps towards a diagrammatic `high-level' alternative for the Hilbert space formalism, one which appeals to our intuition. It allows for intuitive reasoning about interacting quantum systems, and trivialises many otherwise involved and tedious computations. It clearly exposes limitations such as the no-cloning theorem, and phenomena such as quantum teleportation. As a logic, it supports `automation'. It allows for a wider variety of underlying theories, and can be easily modified, having the potential to provide the required step-stone towards a deeper conceptual understanding of quantum theory, as well as its unification with other physical theories. Specific applications discussed here are purely diagrammatic proofs of several quantum computational schemes, as well as an analysis of the structural origin of quantum non-locality. The underlying mathematical foundation of this high-level diagrammatic formalism relies on so-called monoidal categories, a product of a fairly recent development in mathematics. These monoidal categories do not only provide a natural foundation for physical theories, but also for proof theory, logic, programming languages, biology, cooking, ... The challenge is to discover the necessary additional pieces of structure that allow us to predict genuine quantum phenomena.

arXiv comments
Commissioned paper for Contemporary Physics, 31 pages, 84 pictures, some color
Journal reference
Contemporary Physics 51, 59-83, 2010.
DOI
doi:10.1080/00107510903257624
Subjects
quant-phQuantum Physics math-phMathematical Physics math.CTCategory Theory math.QAQuantum Algebra

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  1. A particle physics course for high-school students 1 comment

    • Quantum Picturalism: Learning Quantum Theory in High School https://arxiv.org/pdf/2312.03653 seems a longer version. That QPict Quantum picturalism (2009) paper is https://arxiv.org/pdf/0908.1787 . Picturing quantum processes - A first course in quantum theory and diagrammatic…

      mncharityView on HNParent context

  2. Kindergarten Quantum Mechanics 2 comments

    • From [0] 2018: "In [1] we present an entirely diagrammatic presentation of quantum the-ory with applications in quantum foundations and quantum information.This was the result of many years of work by many, and started of as acategory-theoretic axiomatisation motivated by…

      mncharityView on HNParent context

    • Apparently there is an actual experiment on six-year-olds taking place [1,2]: "Experiment. Consider ten children of ages between six and ten and consider ten high-school teachers of physics and mathematics. The high-school teachers of physics and mathematics will have all the…

      mathgeniusView on HNParent context

  3. Have you ever asked yourself “how did research get done before LateX?” 1 comment

    • I believe Roger Penrose draws all his own illustrations. For copius excellent examples, see his book Road to Reality. Here is one example: https://phys.org/news/2018-03-math-bridges-holography-twisto... https://theportal.wiki/images/1/11/Penrose-Rindler-Clifford-... He also…

      mikhailfrancoView on HNParent context

  4. A brief introduction to quantum computing 1 comment

    • Quantum computing is all about the tensor product. This is, roughly, how "and" works in the quantum world. Learning about tensor products by calculating with matrices is dizzyingly complicated [1]. But there's another way, using string diagrams [2]. [1]…

      mathgeniusView on HNParent context

  5. The Wholeness of Quantum Reality: An Interview with Physicist Basil Hiley 1 comment

    • I think it is generally useful for people to have a picture in their head about whats going on as work quite frequently gets lead by instinct and educated guesswork which can be made rigorous once you've decided its worth the time to do so. That said I really don't understand…

      uselessView on HNParent context

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