categorical quantum mechanics

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 dagger compact category allows one to distinguish between an "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.{{cite book |author1-link=Samson Abramsky |first1=Samson |last1=Abramsky |author2-link=Bob Coecke |first2=Bob |last2=Coecke |arxiv=quant-ph/0402130 |chapter=A categorical semantics of quantum protocols |chapter-url= |editor= |title=Proceedings of the 19th IEEE conference on Logic in Computer Science (LiCS'04) |publisher=IEEE |location= |date=2004 |isbn= |pages= |url=}} In quantum theory, it being compact closed is related to the Choi-Jamiołkowski isomorphism (also known as process-state duality), while the dagger structure captures the ability to take adjoints of linear maps.
• Considering only the morphisms that are completely positive maps, one can also handle mixed states, allowing the study of quantum channels categorically.{{cite book |first=P. |last=Selinger |chapter=Dagger compact closed categories and completely positive maps |chapter-url=http://www.mscs.dal.ca/~selinger/papers.html#dagger |editor= |title=Proceedings of the 3rd International Workshop on Quantum Programming Languages, Chicago, June 30–July 1 |publisher= |location= |date=2005 |isbn= |pages= |url=}}
• 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.{{cite book |first1=B. |last1=Coecke |first2=D. |last2=Pavlovic |arxiv=quant-ph/0608035 |chapter=16. Quantum measurements without sums §16.2 Categorial Semantics |chapter-url={{GBurl|kd8R_Y10U3oC|p=567}} |title=Mathematics of Quantum Computing and Technology |publisher=Taylor and Francis |location= |date=2007 |isbn=9781584889007 |pages=567–604 |url=}}
• In early works, dagger biproducts were used to study both classical communication and the superposition principle. Later, these two features have been separated.{{cite book |first1=B. |last1=Coecke |first2=S. |last2=Perdrix |arxiv=1004.1598|chapter=Environment and classical channels in categorical quantum mechanics|chapter-url=|editor=|title=Proceedings of the 19th EACSL Annual Conference on Computer Science Logic (CSL) |publisher=Springer |series=Lecture Notes in Computer Science |year=2012 |volume=6247 |issue=4 |isbn= |pages= |doi=10.2168/LMCS-8(4:14)2012 |s2cid=16833406|url=}}
• Complementary Frobenius algebras embody the principle of complementarity, which is used to great effect in quantum computation, as in the ZX-calculus.{{cite book |first1=B. |last1=Coecke |first2=R. |last2=Duncan |arxiv=0906.4725 |chapter=Interacting quantum observables |chapter-url= |editor= |title=Proceedings of the 35th International Colloquium on Automata, Languages and Programming (ICALP) |series=Lecture Notes in Computer Science |year=2011 |volume=5126 |isbn= |pages=298–310 |doi=10.1088/1367-2630/13/4/043016 |s2cid=14259278 |url=}}

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,{{cite journal|first1=G.M.|last1=Kelly |first2=M.L. |last2=Laplaza |title=Coherence for compact closed categories |journal=Journal of Pure and Applied Algebra |volume=19 |issue= |pages=193–213|date=1980|doi= 10.1016/0022-4049(80)90101-2|url=|doi-access=free }} Andre Joyal and Ross Street,{{cite journal|first1=A.|last1=Joyal|authorlink1=Andre Joyal|first2=R.|last2=Street | authorlink2=Ross Street |title=The Geometry of tensor calculus I|journal=Advances in Mathematics|volume=88|issue=1 |pages=55–112 |date=1991 |doi= 10.1016/0001-8708(91)90003-P|doi-access=free}} A. Carboni and R. F. C. Walters,{{cite journal |first1=A. |last1=Carboni|first2=R.F.C. |last2=Walters |title=Cartesian bicategories I |journal=Journal of Pure and Applied Algebra|volume=49|issue= 1–2|pages=11–32|date=1987|doi= 10.1016/0022-4049(87)90121-6|url=|doi-access=free }} and

Steve Lack.{{cite journal|first=S. |last=Lack |title=Composing PROPs|journal=Theory and Applications of Categories |volume=13|issue=|pages=147–163|date=2004|doi=|url=}}

Modern textbooks include Categories for quantum theory{{cite book |first1=C. |last1=Heunen |first2=J. |last2=Vicary|title=Categories for Quantum Theory |publisher=Oxford University Press |date=2019 |isbn=978-0-19-873961-6 |pages= |url=https://global.oup.com/academic/product/categories-for-quantum-theory-9780198739616}} and Picturing quantum processes.{{cite book |first1=B. |last1=Coecke |first2=A. |last2=Kissinger |title=Picturing Quantum Processes|publisher=Cambridge University Press|date=2017|isbn=978-1-107-10422-8 |bibcode=2017pqp..book.....C |url={{GBurl|I9gcDgAAQBAJ|pg=PR5}}}}

One of the most notable features of categorical quantum mechanics is that the compositional structure can be faithfully captured by string diagrams.{{cite journal |first=B. |last=Coecke |title=Quantum picturalism |arxiv=0908.1787 |journal=Contemporary Physics |volume=51 |issue= 1|pages=59–83 |date=2010 |doi= 10.1080/00107510903257624|bibcode=2010ConPh..51...59C |s2cid=752173 |url=}}

{{wide image|Teleport.png|780px|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.{{cite book |first=R. |last=Penrose |chapter=Applications of negative dimensional tensors |chapter-url= |editor-first=D. |editor-last=Welsh |title=Combinatorial Mathematics and its Applications. Proceedings of a Conference Held at the Mathematical Institute, Oxford, from 7-10 July, 1969 |publisher=Academic Press |oclc=257806578

|date=1971 |isbn= |pages=221–244 |url=}} 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.{{Cite journal|last=Backens|first=Miriam|date=2014|title=The ZX-calculus is complete for stabilizer quantum mechanics|url=http://stacks.iop.org/1367-2630/16/i=9/a=093021|journal=New Journal of Physics|language=en|volume=16|issue=9|pages=093021|doi=10.1088/1367-2630/16/9/093021|issn=1367-2630|arxiv=1307.7025|bibcode=2014NJPh...16i3021B|s2cid=27558474}} 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.{{cite arXiv|last1=Jeandel|first1=Emmanuel|last2=Perdrix|first2=Simon|last3=Vilmart|first3=Renaud|date=2017-05-31|title=A Complete Axiomatisation of the ZX-Calculus for Clifford+T Quantum Mechanics|eprint=1705.11151|class=quant-ph}} The ZX-calculus has been used to study for instance measurement-based quantum computing.

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.{{cite book |first=J.C. |last=Baez |chapter=Quantum quandaries: a category-theoretic perspective |chapter-url={{GBurl|AsESDAAAQBAJ|p=240}} |arxiv=quant-ph/0404040 |editor1-first=D. |editor1-last=Rickles |editor2-first=S. |editor2-last=French |editor31-first=J.T. |editor31-last=Saatsi |title=The Structural Foundations of Quantum Gravity |publisher=Oxford University Press |location= |date=2004 |isbn=978-0-19-926969-3 |pages=240–266 |url=}}

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.{{cite journal |first=P. |last=Selinger |title=Finite dimensional Hilbert spaces are complete for dagger compact closed categories |journal=Electronic Notes in Theoretical Computer Science |volume=270 |issue=1 |pages=113–9 |date=2011 |doi=10.1016/j.entcs.2011.01.010 |citeseerx=10.1.1.749.4436 |url=http://www.mscs.dal.ca/~selinger/papers.html#finhilb}}
• Dagger commutative Frobenius algebras in the category of finite-dimensional Hilbert spaces correspond to orthogonal bases.{{cite journal |first1=B. |last1=Coecke |first2=D. |last2=Pavlovic |first3=J. |last3=Vicary |arxiv=0810.0812 |title=A new description of orthogonal bases |journal=Mathematical Structures in Computer Science |volume=23 |issue=3 |pages=555–567 |date=2013 |doi=10.1017/S0960129512000047 |citeseerx=10.1.1.244.6490 |s2cid=12608889 |url=}} A version of this correspondence also holds in arbitrary dimension.{{cite book |first1=S. |last1=Abramsky |first2=C. |last2=Heunen |arxiv=1011.6123 |chapter=H*-algebras and nonunital Frobenius algebras: first steps in infinite-dimensional categorical quantum mechanics |chapter-url= |editor= |title=Clifford Lectures, AMS Proceedings of Symposia in Applied Mathematics |year=2010 |publisher= |location= |isbn= |pages= |url=}} to appear (2010).
• 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.{{cite journal |first=J. |last=Vicary |arxiv=0807.2927 |title=Completeness of dagger-categories and the complex numbers |journal=Journal of Mathematical Physics |volume=52 |issue= 8|pages=082104 |date=2011 |doi=10.1063/1.3549117 |bibcode=2011JMP....52h2104V |s2cid=115154127 |url=}}
• 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.{{cite journal |first=C. |last=Heunen |title=An embedding theorem for Hilbert categories |journal=Theory and Applications of Categories |volume=22 |issue= |pages=321–344 |date=2008 |url= |arxiv=0811.1448 }}
• Six axioms characterize the category of Hilbert spaces completely, fulfilling the reconstruction programme.{{cite journal |first1=C. |last1=Heunen |first2=A. |last2=Kornell |arxiv=2109.07418 |title=Axioms for the category of Hilbert spaces |journal=Proceedings of the National Academy of Sciences |volume=119 |issue=9 |pages=e2117024119 |date=2022 |doi= 10.1073/pnas.2117024119|doi-access=free |pmid=35217613 |pmc=8892366 |bibcode=2022PNAS..11917024H |url=}} Two of these axioms concern a dagger and a tensor product, a third concerns biproducts.
• Special dagger commutative Frobenius algebras in the category of sets and relations correspond to discrete abelian groupoids.{{cite book |first=D. |last=Pavlovic |title=Quantum Interaction. QI 2009 |chapter=Quantum and classical structures in nondeterminstic computation |arxiv=0812.2266 |publisher=Springer |series=Lecture Notes in Computer Science |volume=5494 |date=2009 |doi=10.1007/978-3-642-00834-4_13 |isbn=978-3-642-00834-4 |pages=143–157 |s2cid=11615031 |url=}}(2009).
• Finding complementary basis structures in the category of sets and relations corresponds to solving combinatorical problems involving Latin squares.{{cite arXiv |first1=J. |last1=Evans |first2=R. |last2=Duncan |first3=A. |last3=Lang |first4=P. |last4=Panangaden |title=Classifying all mutually unbiased bases in Rel |date=2009 |class=quant-ph |eprint=0909.4453 }}
• 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.{{cite book |first1=B. |last1=Coecke |first2=A. |last2=Kissinger |arxiv=1002.2540 |chapter=The compositional structure of multipartite quantum entanglement |chapter-url= |editor= |title=Proceedings of the 37th International Colloquium on Automata, Languages and Programming (ICALP) |publisher=Springer |series=Lecture Notes in Computer Science |volume=6199 |date=2010 |isbn= |pages=297–308 |url=}}

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.{{cite thesis |first=R. |last=Duncan |title=Types for Quantum Computing |date=2006 |type=PhD |publisher=University of Oxford |url=http://www.cs.ox.ac.uk/files/2425/Types%20for%20Quantum%20Computation.pdf |citeseerx=10.1.1.122.134 |id=uk.bl.ethos.483690}} There exists [https://quantomatic.github.io/ 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.{{cite journal |first1=C. |last1=Heunen |first2=B. |last2=Jacobs |arxiv=0902.2355 |title=Quantum logic in dagger kernel categories |journal=Order |volume=27 |issue= 2|pages=177–212 |date=2009 |doi= 10.1007/s11083-010-9145-5|s2cid=2760251 |url=}}{{cite journal |first=J. |last=Harding |title=A Link between quantum logic and categorical quantum mechanics |journal=International Journal of Theoretical Physics |volume=48 |issue= 3|pages=769–802 |date=2009 |doi=10.1007/s10773-008-9853-4 |bibcode=2009IJTP...48..769H |citeseerx=10.1.1.605.9683 |s2cid=13394885 |url=}} In fact, the former setting allows logical quantifiers, the existence of which was never satisfactorily addressed in traditional quantum logic.

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.{{cite journal |first1=B. |last1=Coecke |first2=B. |last2=Edwards |first3=R.W. |last3=Spekkens |arxiv=1003.5005 |title=Phase groups and the origin of non-locality for qubits |journal=Electronic Notes in Theoretical Computer Science |volume= 270|issue= 2|pages= 15–36|doi= 10.1016/j.entcs.2011.01.021|url= |year=2011|s2cid=27998267 }}, to appear (2010).

The DisCoCat framework applies categorical quantum mechanics to natural language processing.{{cite arXiv |last1=Coecke |first1=Bob |last2=Sadrzadeh |first2=Mehrnoosh |last3=Clark |first3=Stephen |date=2010-03-23 |title=Mathematical Foundations for a Compositional Distributional Model of Meaning |class=cs.CL |eprint=1003.4394 }} The types of a pregroup grammar are interpreted as quantum systems, i.e. as objects of a dagger compact category. The grammatical derivations are interpreted as quantum processes, e.g. a transitive verb takes its subject and object as input and produces a sentence as output. Function words such as determiners, prepositions, relative pronouns, coordinators, etc. can be modeled using the same Frobenius algebras that model classical communication.{{Cite journal |last1=Sadrzadeh |first1=Mehrnoosh |last2=Clark |first2=Stephen |last3=Coecke |first3=Bob |date=2013-12-01 |title=The Frobenius anatomy of word meanings I: subject and object relative pronouns |arxiv=1404.5278 |journal=Journal of Logic and Computation |volume=23 |issue=6 |pages=1293–1317 |doi=10.1093/logcom/ext044 |issn=0955-792X}}{{cite journal |last1=Sadrzadeh |first1=Mehrnoosh |last2=Clark |first2=Stephen |last3=Coecke |first3=Bob |title=The Frobenius anatomy of word meanings II: possessive relative pronouns |journal=Journal of Logic and Computation |year=2016 |volume=26 |issue=2 |pages=785–815 |arxiv=1406.4690 |doi=10.1093/logcom/exu027}} This can be understood as a monoidal functor from grammar to quantum processes, a formal analogy which led to the development of quantum natural language processing.{{cite arXiv |last1=Coecke |first1=Bob |last2=de Felice |first2=Giovanni |last3=Meichanetzidis |first3=Konstantinos |last4=Toumi |first4=Alexis |date=2020-12-07 |title=Foundations for Near-Term Quantum Natural Language Processing |class=quant-ph |eprint=2012.03755 }}

• ZX-calculus
• String diagram
• Applied category theory
• Quantum foundations

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Category:Quantum mechanics

Category:Category theory

Category:Dagger categories

Category:Monoidal categories