Monoidal Categories for Quantum Theory

Intersession 2026
Illinois Mathematics and Science Academy

Instructor: Dheeran E. Wiggins
Email: dwiggins@imsa.edu or dheeran2@illinois.edu
Dates: January ?? -- January ??, 2026
Room: ??

Welcome to Monoidal Categories for Quantum Theory! We will visit

  • the rudiments of quantum information.
  • Hilbert space basics.
  • (braided, symmetric) monoidal categories \((\mathcal{A},\otimes,\mathbb{1})\).
  • biproducts \(\oplus\) and daggers \(\dagger\) for quantum measurement.
  • duality for quantum teleportation.
  • the graphical calculus for categories.
    • Notably, we will take a rigorous approach to the relevant mathematics, while supplementing the theorems with physical and information-theoretic intuition.

    Studying quantum (information) theory from a categorical perspective requires working through many layers of abstraction. No matter how comfortable you are with any of the relevant mathematics or physics, it will be hard. My hope is that by the end of these lectures, you will be well-equipped to break into the literature, whether textbooks or papers. In particular, on your homework I will point you to papers that make use of these tools.

    I will tweak the schedule based on interest and the level of mathematical background of the class. Ask questions and follow along as best you can. Intersession is a chance to learn something neat outside the stress of your standard coursework.

    Textbook

    No textbook is required. We will follow chapters 0 through 3 of Categories for Quantum Theory: An Introduction by Chris Heunen and Jamie Vicary, though we may deviate depending on time and interest. The following are also good references.

  • Introduction to Categorical Quantum Mechanics by Chris Heunen and Jamie Vicary. These are the wonderful notes that were later adapted ino the above book.
  • Quantum Quandaries: a Category-Theoretic Perspective by John Baez. I appreciated this (well-known) piece of writing, which motivates some categorical notions in quantum theory, when I first began learning the subject. You may too.
  • Category Theory in Context by Emly Riehl. This is without a doubt my favorite introductory text on categories.
  • Handbook of Categorical Algebra by Francis Borceux. If there are any holes in your knowledge of categorical structures of any kind, one of Borceux's three volumes is bound to have it, or something analogous that points you in a worthwhile direction.

    • Assignments and Deliverables

    There will be three to four exercises assigned to be done as homework between lecture days. These will be pretty brief and should be easy after sitting through lecture. If not, I am happy to work through them with you! At times, students will be asked to present their solutions to the class.

    We will also develop an expositional \(\LaTeX\) poster together, to be presented at the Intersession Expo, summarizing memorable theorems from the week.

    Catalog Description

    Introduced into physics literature by Roger Penrose in the early 1970s to simplify the indexing of tensor calculations, the Penrose graphical calculus has become a compelling diagrammatic tool for contemporary quantum theory. As it turns out, Penrose’s notation is an instance of a wider class of graphical calculi associated with monoidal categories. Where categories generalize the mathematical notions of sets and functions, monoidal categories, in part, generalize multiplication of natural numbers to multiplication of structures. Notably, a variety of monoidal categories naturally encodes the operation of tensoring Hilbert spaces, and thus of combining quantum systems. After introducing the rudiments of category theory and quantum computation, we will study states, effects, superposition, and measurement from the formal perspective of symmetric monoidal dagger categories. In particular, we will emphasize the graphical calculus associated with these categories, which will prepare students for using Penrose notation or the ZX calculus.

    Acknowledgements

    This intersession would not exist without

  • Anderson Trimm, who gave me my first introduction to category theory in an intersession of his own.
  • Charles Rezk, whose mentorship while I studied (symmetric, rigid, dagger, blah, blah) monoidal categories was indispensable.
  • Felix Leditzky and Eric Chitambar, who showed me just how exciting mathematical problems in QIT can be.
  • Roy Araiza and Jihong Cai, from whom I learned so much mathematics, quantum and otherwise.