Quantum picturalism

By Bob Coecke. 2009

247 citations recorded by the source corpus.

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.

Subfields: Artificial Intelligence, Atomic and Molecular Physics, and Optics

View Paper: https://www.semanticscholar.org/paper/45848c91efa41e1251b8b90f66121d4bf1e0c569

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In Artificial Intelligence

In Atomic and Molecular Physics, and Optics

Citations in the corpus

Reformulating and Reconstructing Quantum Theory (2011, background): “One of the most salient features of the categorical approach is that it gives rise to a kind of pictorialism [16].”
A mathematical theory of resources (2014, unknown): “Note that one can just as easily deﬁne a resource theory of asymmetry for classical process theories or indeed for any process theory, including cases that are neither classical nor quantum such as the framework of generalized operational theories [17, 15].”
Mathematical Foundations for a Compositional Distributional Model of Meaning (2010, background): “Note that the above pictures are very similar to the ones encountered in [7, 9] which describe quantum informatic protocols such as quantum teleportation and entanglement swapping.”
Interacting quantum observables: categorical algebra and diagrammatics (2009, background): “A more detailed presentation of the physical intuition behind smcs can be found in [14, 24, 17]; [24] is an extensive tutorial specifically written to provide the appropriate background on the kind of category that is required for this paper.”