Quantum Process Theories

Origin. Abramsky and Coecke (2004). Categorical quantum mechanics. Processes as morphisms. Dagger compact categories. Foundation for quantum protocols.

Models. Systems as objects. Processes as morphisms. Sequential composition: g ∘ f. Parallel: f ⊗ g. Dagger: time reversal. Compactness: cups and caps.

Formalism.

Dagger category: †: C^op → C functor f: A → B gives f†: B → A (f†)† = f (g ∘ f)† = f† ∘ g†

Monoidal structure: ⊗: C × C → C (tensor) I: unit object Associativity, unit coherence.

Compact closed: Every object A has dual A*. η: I → A* ⊗ A (cup) ε: A ⊗ A* → I (cap) Yanking: (ε ⊗ id) ∘ (id ⊗ η) = id

Dagger compact: Dagger + compact + compatibility. (η_A)† = ε_A (cups and caps dual)

Quantum interpretation: Objects: Hilbert spaces Morphisms: linear maps Dagger: adjoint Tensor: tensor product Compact: finite dimensional

Graphical calculus: String diagrams. Wires: systems. Boxes: processes. Topology captures equations.

CPM construction: Completely positive maps. Quantum channels. Environment as purification.

Symbols.

Symbol	Unicode	Name	Meaning
†	U+2020	Dagger	Adjoint/reversal
⊗	U+2297	Tensor	Parallel
∘	U+2218	Compose	Sequential
*	—	Dual	Dual object

Metatheory. Coherence theorems. Graphical soundness. Completeness for relations. CPM construction.

Applies to. Quantum computing. Protocol design. Quantum foundations. Compositional semantics.

Limitations. Abstract. Category theory needed. Infinite dimensions harder. Physics abstraction.