Quantum picturalism

Abstract

Why did it take us 50 years since the birth of the quantum mechanical formalism to discover that unknown quantum states cannot be cloned? Yet, the proof of the ‘no-cloning theorem’ is easy, and its consequences and potential for applications are immense. Similarly, why did it take us 60 years to discover the conceptually intriguing and easily derivable physical phenomenon of ‘quantum teleportation’? We claim that 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. Using a technical term from computer science, the quantum mechanical formalism is ‘low-level’. In this review we present steps towards a diagrammatic ‘high-level’ alternative for the Hilbert space formalism, one which appeals to our intuition.

The diagrammatic language as it currently stands 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 enables a (classical) computer to reason about interacting quantum systems, prove theorems, and design protocols. 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. Its logical underpinning is linear logic, an even more recent product of research in logic and computer science. These monoidal categories do not only provide a natural foundation for physical theories, but also for proof theory, logic, programming languages, biology, cooking, … So the challenge is to discover the necessary additional pieces of structure that allow us to predict genuine quantum phenomena. These additional pieces of structure represent the capabilities nature has provided us with to manipulate entities subject to the laws of quantum theory.

Keywords:

• diagrammatic reasoning
• quantum information and computation
• quantum foundations
• monoidal categories and linear logic
• axiomatic quantum theory

Acknowledgements

We are grateful for constructive critical feedback by Andreas Doering, Lucien Hardy, Basil Hiley, Chris Isham, Terry Rudolph and Rob Spekkens which affected our presentation here. We thank Jacob Biamonte, Peter Morgan, Rob Spekkens, James Whitfield and the anonymous reviewers for carefully reading this paper and providing appropriate feedback. The author is supported by an EPSRC Advanced Research Fellowship entitled ‘The Structure of Quantum Information and its Ramifications for IT’, by an FQXi Large Grant entitled ‘The Road to a New Quantum Formalism: Categories as a Canvas for Quantum Foundations’, and by an EC-FP6-STREP entitled ‘Foundational Structures in Quantum Information and Computation’.

Notes

1. This presentation was directly taken from the report of one of the referees, since I couldn't put it better myself.

2. The notation x: = y stands for ‘ x is defined to be y’. One doesn't lose much if one reads this as an ordinary equality.

3. That we have to rotate the kets and bras is merely a consequence of our convention to read the pictures from bottom to top with respect to composition. In other words, in our pictures time flows upward.

4. There are of course several important subtleties when thinking of Hilb or FHilb as modelling quantum processes. For example, the states of a quantum system are not described by vectors in a Hilbert space but rather by one-dimensional subspaces. The categorical formalism can easily handle this 74, but a detailed discussion is beyond the scope of this paper.

5. So the syntax f : X → Y :: x ↦ y that we use to denote functions consists of two parts. The part X → Y tells us that X is the set of arguments and that the function takes values in Y. The part x ↦ y tells us that f(x): = y.

6. Depending on one's taste one can depict these either as or as ; here we picked the latter.

7. Following Atiyah in 75, topological quantum field theories can be succinctly defined as monoidal functors from nCob into FHilb, where a functor is a map both on objects and on morphisms which preserves composition and tensor 22,23,76.