A Categorical Semantics of Quantum Protocols
A Categorical Semantics of Quantum Protocols
Abramsky and Coecke give a compositional semantics for quantum protocols using compact closed categories with biproducts.
Paper information (2004, LICS) — Samson Abramsky and Bob Coecke. DOI; arXiv
Work
The paper recasts the axiomatic structure of quantum information and computation in compact closed categories with biproducts. It uses this abstract structure to represent protocols such as teleportation, gate teleportation, and entanglement swapping, placing states, processes, classical branching, and information flow in one compositional picture. It also explains how scalars can supply the probabilities required by the Born rule, connecting qualitative categorical structure with quantum measurement.
Contributions
- An early categorical quantum-mechanics framework for expressing quantum protocols through compact structure and biproducts.
- Compositional accounts of teleportation, gate teleportation, and entanglement swapping.
- An explanation of how scalars and the Born rule emerge from the abstract structure, including requirements on the scalar semiring for protocols.
- A single formal setting for quantum information flow and branching caused by quantum indeterminacy.
Limits
Categorical models are strong at expressing compositional structure and protocol reasoning, but a concrete category and scalar interpretation are needed to recover standard Hilbert-space behavior or a program implementation. The paper does not provide hardware compilation, noise calibration, complexity estimates, or a type checker for a particular quantum language. An abstractly expressible protocol need not have an efficient implementation or a machine-checked engineering proof.
Connections
The paper complements Selinger's QPL language: QPL focuses on writable program constructs and superoperator semantics, while this work focuses on abstract compositional protocol structure. It can also be compared with van Tonder's quantum lambda calculus, which starts from syntax and equational reasoning. Later work on dagger compact categories, the CPM construction, and graphical languages extends the pure-protocol semantics developed here.
Original paper
Samson Abramsky and Bob Coecke, A Categorical Semantics of Quantum Protocols, in LICS 2004, DOI: 10.1109/LICS.2004.1319636. arXiv:quant-ph/0402130