Categorical quantum mechanics

Categorical quantum mechanics is the study of quantum foundations and quantum information using paradigms from mathematics and computer science, notably monoidal category theory. The primitive objects of study are physical processes, and the different ways that these can be composed. It was pioneered in 2004 by Abramsky and Coecke.

Mathematical setup

Mathematically, the basic setup is captured by a dagger symmetric monoidal category: composition of morphisms models sequential composition of processes, and the tensor product describes parallel composition of processes. The role of the dagger is to assign to each state a corresponding test. These can then be adorned with more structure to study various aspects, including:

• A dagger compact category allows one to distinguish between "input" and "output" of a process. In the diagrammatic calculus, it allows wires to be bent, allowing for a less restricted transfer of information. In particular, it allows entangled states and measurements, and gives elegant descriptions of protocols such as quantum teleportation.^[1]
• Considering only the morphisms that are completely positive maps, one can also handle mixed states, allowing the study of quantum channels categorically.^[2]
• Wires are always two-ended (and can never be split into a Y), reflecting the no-cloning and no-deleting theorems of quantum mechanics.
• Special commutative dagger Frobenius algebras model the fact that certain processes yield classical information, that can be cloned or deleted, thus capturing classical communication.^[3]
• In early works, dagger biproducts were used to study both classical communication and the superposition principle. Later, these two features have been separated.^[4]
• Complementary Frobenius algebras embody the principle of complementarity, which is used to great effect in quantum computation.^[5]

A substantial portion of the mathematical backbone to this approach is drawn from Australian category theory, most notably from work by Kelly and Laplaza,^[6] Joyal and Street,^[7] Carboni and Walters,^[8] and Lack.^[9]

Diagrammatic calculus

One of the most notable features of categorical quantum mechanics is that the compositional structure can be faithfully captured by a purely diagrammatic calculus.^[10]

An illustration of the diagrammatic calculus: the quantum teleportation protocol as modeled in categorical quantum mechanics.

These diagrammatic languages can be traced back to Penrose graphical notation, developed in the early 1970s.^[11]

Branches of activity

Axiomatization and new models

One of the main successes of the categorical quantum mechanics research program is the sheer ratio of results to assumptions. From seemingly very weak abstract constraints on the compositional structure it was possible to derive many quantum mechanical phenomena. In contrast to earlier axiomatic approaches which aimed to reconstruct Hilbert space quantum theory from reasonable assumptions, this attitude of not aiming for a complete axiomatization may lead to new interesting models that describe quantum phenomena, which could be of use when crafting the theories of the future.^[12]

Completeness and representation results

There are several theorems relating the abstract setting of categorical quantum mechanics to traditional settings for quantum mechanics:

• Completeness of the diagrammatic calculus: an equality of morphisms can be proved in the category of finite-dimensional Hilbert spaces if and only if it can be proved in the graphical language of dagger compact closed categories.^[13]
• Dagger commutative Frobenius algebras in the category of finite-dimensional Hilbert spaces correspond to orthogonal bases.^[14] This can be extended to arbitrary dimensions.^[15]
• Certain extra axioms guarantee that the scalars embed into the field of complex numbers, namely the existence of finite dagger biproducts and dagger equalizers, well-pointedness, and a cardinality restriction on the scalars.^[16]
• Certain extra axioms on top of the previous guarantee that a dagger symmetric monoidal category embeds into the category of Hilbert spaces, namely if every dagger monic is a dagger kernel. In that case the scalars in fact form an involutive field instead of just embedding in one. If the category is compact, the embedding lands in finite-dimensional Hilbert spaces.^[17]
• Special dagger commutative Frobenius algebras in the category of sets and relations correspond to discrete Abelian groupoids.^[18]
• Finding complementary basis structures in the category of sets and relations corresponds to solving combinatorical problems involving Latin squares.^[19]
• Dagger commutative Frobenius algebras on qubits must be either special or antispecial, relating to the fact that maximally entangled tripartite states are SLOCC-equivalent to either the GHZ or the W state.^[20]

Categorical quantum mechanics as logic

Categorical quantum mechanics can also be seen as a type theoretic form of quantum logic that, in contrast to traditional quantum logic, supports formal deductive reasoning.^[21] There exists software that supports and automates this reasoning.

There is another connection between categorical quantum mechanics and quantum logic: subobjects in certain dagger categories form orthomodular lattices, namely in dagger kernel categories^[22] and dagger complemented biproduct categories.^[23] In fact, the former setting enables logical quantifiers, which problem was never satisfactorily addressed in traditional quantum logic, but becomes clear through a categorical approach.

Categorical quantum mechanics as a high-level approach to quantum information and computation

Categorical quantum mechanics, when applied to quantum information theory or quantum computing, provides high-level methods for these areas. For example, Measurement Based Quantum Computing.

Categorical quantum mechanics as foundation for quantum mechanics

The framework can be used to describe theories more general than quantum theory. This enables one to study which features single out quantum theory in contrast to other non-physical theories, and this may provide important insights in the nature of quantum theory. For example, the framework is flexible enough to provide a succinct compositional description of Spekkens' Toy Theory and enabled to pinpoint which structural ingredient causes it to be different from quantum theory.^[24]

Notes

References

1. Samson Abramsky and Bob Coecke, A categorical semantics of quantum protocols, Proceedings of the 19th IEEE conference on Logic in Computer Science (LiCS'04). IEEE Computer Science Press (2004).
2. P. Selinger, Dagger compact closed categories and completely positive maps, Proceedings of the 3rd International Workshop on Quantum Programming Languages, Chicago, June 30–July 1 (2005).
3. B. Coecke and D. Pavlovic, Quantum measurements without sums. In: Mathematics of Quantum Computing and Technology, pages 567–604, Taylor and Francis (2007).
4. B. Coecke and S. Perdrix, Environment and classical channels in categorical quantum mechanicsIn: Proceedings of the 19th EACSL Annual Conference on Computer Science Logic (CSL), Lecture Notes in Computer Science 6247, Springer-Verlag.
5. B. Coecke and R. Duncan, Interacting quantum observables In: Proceedings of the 35th International Colloquium on Automata, Languages and Programming (ICALP), pp. 298–310, Lecture Notes in Computer Science 5126, Springer.
6. G.M. Kelly and M.L. Laplaza, Coherence for compact closed categories, Journal of Pure and Applied Algebra 19, 193–213 (1980).
7. A. Joyal and R. Street, The Geometry of tensor calculus I, Advances in Mathematics 88, 55–112 (1991).
8. A. Carboni and R. F. C. Walters, Cartesian bicategories I, Journal of Pure and Applied Algebra 49, 11–32 (1987).
9. S. Lack, Composing PROPs, Theory and Applications of Categories 13, 147–163 (2004).
10. B. Coecke, Quantum picturalism, Contemporary Physics 51, 59–83 (2010).
11. R. Penrose, Applications of negative dimensional tensors, In: Combinatorial Mathematics and its Applications, D.~Welsh (Ed), pages 221–244. Academic Press (1971).
12. J. C. Baez, Quantum quandaries: a category-theoretic perspective. In: The Structural Foundations of Quantum Gravity, D. Rickles, S. French and J. T. Saatsi (Eds), pages 240–266. Oxford University Press (2004).
13. P. Selinger, Finite dimensional Hilbert spaces are complete for dagger compact closed categories. Electronic Notes in Theoretical Computer Science, to appear (2010).
14. B. Coecke, D. Pavlovic, and J. Vicary, A new description of orthogonal bases. Mathematical Structures in Computer Science, to appear (2008).
15. S. Abramsky and C. Heunen H*-algebras and nonunital Frobenius algebras: first steps in infinite-dimensional categorical quantum mechanics, Clifford Lectures, AMS Proceedings of Symposia in Applied Mathematics, to appear (2010).
16. J. Vicary, Completeness of dagger-categories and the complex numbers, Journal of Mathematical Physics, to appear (2008).
17. C. Heunen, An embedding theorem for Hilbert categories. Theory and Applications of Categories 22, 321–344. (2008)
18. D. Pavlovic, Quantum and classical structures in nondeterminstic computation, Lecture Notes in Computer Science 5494, page 143–157, Springer (2009).
19. J. Evans, R. Duncan, A. Lang and P. Panangaden, Classifying all mutually unbiased bases in Rel (2009).
20. B. Coecke and A. Kissinger The compositional structure of multipartite quantum entanglement, Proceedings of the 37th International Colloquium on Automata, Languages and Programming (ICALP), pages 297–308, Lecture Notes in Computer Science 6199, Springer (2010).
21. R. Duncan (2006) Types for Quantum Computing, DPhil. thesis. University of Oxford.
22. C. Heunen and B. Jacobs, Quantum logic in dagger kernel categories. Order 27, 177–212 (2009).
23. J. Harding, A Link between quantum logic and categorical quantum mechanics, International Journal of Theoretical Physics 48, 769–802 (2009).
24. B. Coecke, B. Edwards and R. W. Spekkens, Phase groups and the origin of non-locality for qubits, Electronic Notes in Theoretical Computer Science, to appear (2010).