Categories for Quantum Theory: An Introduction

Peter Nonnenmann

Srinjoy Ganguly, Sai Nandan Morapakula, Luis Gerardo Ayala Bertel

Quantum Natural Language Processing (QNLP) is an emerging field of Quantum AI where natural language is combined with quantum computing capabilities. The origins of this field come from an abstract mathematical concept called category theory especially monoidal categories which have a diagrammatic structure. These monoidal categories have been utilized to describe various quantum mechanical phenomena in a purely pictorial fashion. When a model of natural language is equivalent to a model which explains quantum theory it signifies that QNLP is quantum-native. This chapter provides a friendly introduction to QNLP by providing the necessary background details such as—diagrammatic quantum theory and ZX calculus and explains the compositional structure of Lambek’s pregroup grammar because of which QNLP models are now able to run on real quantum hardware.

Kevin Dunne

Andreas Döring

Topos theory, a branch of category theory, has been proposed as mathematical basis for the formulation of physical theories. In this article, we give a brief introduction to this approach, emphasizing the logical aspects. Each topos serves as a ‘mathematical universe’ with an internal logic, which is used to assign truth-values to all propositions about a physical system. We show in detail how this works for (algebraic) quantum theory.

Ruth E. Kastner

This chapter serves as an elementary introduction to the basic quantum puzzles such as the measurement problem, indeterminacy, nonlocality, entanglement, and ambiguity surrounding the projection postulate. It briefly reviews the standard extant interpretations such as the Copenhagen interpretation, many-worlds interpretation, and hidden variables interpretation, and introduces the idea that quantum theory is about a new metaphysical category: physical possibility or res potentia .

Peter Woit