Categorical Quantum Mechanics

Categorical quantum mechanics rebuilds quantum theory from the compositional structure of processes rather than from the Hilbert-space formalism, keeping only what is needed to talk about states, effects, and their composition. It sits inside a categorification ladder that treats the complex numbers themselves as the ground floor of a tower of Hilbert-like structures.

The ladder reads as the one-dimensional Hilbert space and as a one-dimensional -Hilbert space, where consists of finite-dimensional Hilbert spaces and linear maps. Above it sits as the next rung, with a conjectural -Hilbert hierarchy beyond.

At the working level one abstracts the ambient -algebras and quantum processes to a dagger-compact symmetric monoidal category (with -category structure supplying the analytic content), where the dagger models adjoints and compactness supplies the caps and cups that encode entanglement and teleportation.

Definition of scalars as an endomorphism monoid

In any monoidal category the scalars are the endomorphisms of the tensor unit, , and they form a commutative monoid. In a dagger-compact category modelling quantum processes this monoid recovers . Amplitudes, inner products, and the Born rule live entirely in , so numbers re-emerge as a special case of morphisms.

Remark on probabilistic cousins

Replacing the dagger-compact axioms by a copy/discard comonoid on each wire moves from pure quantum processes to a markov-category, the categorical home of classical probability. The two settings share the string-diagrammatic language and differ in which extra structure each wire carries.

References

• Samson Abramsky and Bob Coecke, “A categorical semantics of quantum protocols”, LICS 2004, arXiv:quant-ph/0402130
• John C. Baez, “Higher-Dimensional Algebra II: 2-Hilbert Spaces”, Advances in Mathematics 127 (1997), arXiv:q-alg/9609018