# 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 Samson Abramsky and Bob 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. For instance:

A substantial portion of the mathematical backbone to this approach is drawn from Australian category theory, most notably from work by Max Kelly and M. L. Laplaza,[6] Andre Joyal and Ross Street,[7] A. Carboni and R. F. C. Walters,[8] and Steve Lack.[9] Modern textbooks include [10] and [11].

## 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.[12]

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.[13] Diagrammatic reasoning has been used before in quantum information science in the quantum circuit model, however, in categorical quantum mechanics primitive gates like the CNOT-gate arise as composites of more basic algebras, resulting in a much more compact calculus.[14] In particular, the ZX-calculus has sprung forth from categorical quantum mechanics as a diagrammatic counterpart to conventional linear algebraic reasoning about quantum gates. The ZX-calculus consists of a set of generators representing the common Pauli quantum gates and the Hadamard gate equipped with a set of graphical rewrite rules governing their interaction. Although a standard set of rewrite rules has not yet been established, some versions have been proven to be complete, meaning that any equation that holds between two quantum circuits represented as diagrams can be proven using the rewrite rules.[15] The ZX-calculus has been used to study for instance measurement based quantum computing.

## Branches of activity

### Axiomatization and new models

One of the main successes of the categorical quantum mechanics research program is that from seemingly weak abstract constraints on the compositional structure, it turned out to be 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 future theories.[16]

### 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.[17]
• Dagger commutative Frobenius algebras in the category of finite-dimensional Hilbert spaces correspond to orthogonal bases.[18] A version of this correspondence also holds in arbitrary dimension.[19]
• 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.[20]
• 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 form an involutive field instead of just embedding in one. If the category is compact, the embedding lands in finite-dimensional Hilbert spaces.[21]
• Special dagger commutative Frobenius algebras in the category of sets and relations correspond to discrete abelian groupoids.[22]
• Finding complementary basis structures in the category of sets and relations corresponds to solving combinatorical problems involving Latin squares.[23]
• 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.[24]

### 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.[25] There exists software that supports and automates this reasoning.

There is another connection between categorical quantum mechanics and quantum logic, as subobjects in dagger kernel categories and dagger complemented biproduct categories form orthomodular lattices.[26][27] In fact, the former setting allows logical quantifiers, the existence of which was never satisfactorily addressed in traditional quantum logic.

### Categorical quantum mechanics as foundation for quantum mechanics

Categorical quantum mechanics allows a description of more general theories than quantum theory. This enables one to study which features single out quantum theory in contrast to other non-physical theories, hopefully providing some insight into the nature of quantum theory. For example, the framework allows a succinct compositional description of Spekkens' toy theory that allows one to pinpoint which structural ingredient causes it to be different from quantum theory.[28]

## 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. ^ C. Heunen and J. Vicary, Categories for Quantum Theory, Oxford University Press (2019)
11. ^ B. Coecke and A. Kissinger, Picturing Quantum Processes, Cambridge University Press (2017)
12. ^ B. Coecke, Quantum picturalism, Contemporary Physics 51, 59–83 (2010).
13. ^ R. Penrose, Applications of negative dimensional tensors, In: Combinatorial Mathematics and its Applications, D.~Welsh (Ed), pages 221–244. Academic Press (1971).
14. ^ Backens, Miriam (2014). "The ZX-calculus is complete for stabilizer quantum mechanics". New Journal of Physics. 16 (9): 093021. arXiv:1307.7025. Bibcode:2014NJPh...16i3021B. doi:10.1088/1367-2630/16/9/093021. ISSN 1367-2630.
15. ^ Jeandel, Emmanuel; Perdrix, Simon; Vilmart, Renaud (2017-05-31). "A Complete Axiomatisation of the ZX-Calculus for Clifford+T Quantum Mechanics". arXiv:1705.11151 [quant-ph].
16. ^ 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).
17. ^ P. Selinger, Finite dimensional Hilbert spaces are complete for dagger compact closed categories. Electronic Notes in Theoretical Computer Science, to appear (2010).
18. ^ B. Coecke, D. Pavlovic, and J. Vicary, A new description of orthogonal bases. Mathematical Structures in Computer Science, to appear (2008).
19. ^ 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).
20. ^ J. Vicary, Completeness of dagger-categories and the complex numbers, Journal of Mathematical Physics, to appear (2008).
21. ^ C. Heunen, An embedding theorem for Hilbert categories. Theory and Applications of Categories 22, 321–344. (2008)
22. ^ D. Pavlovic, Quantum and classical structures in nondeterminstic computation, Lecture Notes in Computer Science 5494, page 143–157, Springer (2009).
23. ^ J. Evans, R. Duncan, A. Lang and P. Panangaden, Classifying all mutually unbiased bases in Rel (2009).
24. ^ 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).
25. ^ R. Duncan (2006) Types for Quantum Computing, DPhil. thesis. University of Oxford.
26. ^ C. Heunen and B. Jacobs, Quantum logic in dagger kernel categories. Order 27, 177–212 (2009).
27. ^ J. Harding, A Link between quantum logic and categorical quantum mechanics, International Journal of Theoretical Physics 48, 769–802 (2009).
28. ^ 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).