Categorical quantum mechanics is the study of quantum physics using methods from mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

 and computer science

Computer science

Computer science or computing science is the study of the theoretical foundations of information and computation and of practical techniques for their implementation and application in computer systems...

, notably category theory

Category theory

Category theory is an area of study in mathematics that examines in an abstract way the properties of particular mathematical concepts, by formalising them as collections of objects and arrows , where these collections satisfy certain basic conditions...

, that emphasize compositionality. The primitive objects of study are (physical) processes

Process (science)

In science, a process is every sequence of changes of a real object/body which is observable using the scientific method. Therefore, all sciences analyze and model processes....

, which can be composed both in sequence and in parallel.

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. These can then by adorned with more structure to study various aspects, including:

• A dagger compact category
  Dagger compact category
  In mathematics, dagger compact categories first appeared in 1989 in the work of Doplicher and Roberts on the reconstruction of compact topological group from their category of finite-dimensional continuous unitary representations...
  
   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 to entangled states and measurements, and gives elegant descriptions of protocols such as quantum teleportation
  Quantum teleportation
  Quantum teleportation, or entanglement-assisted teleportation, is a process by which a qubit can be transmitted exactly from one location to another, without the qubit being transmitted through the intervening space...
  
  .
• Considering only the morphisms that are completely positive maps, one can also handle mixed states, allowing the study of quantum channel
  Quantum channel
  In quantum information theory, a quantum channel is a communication channel which can transmit quantum information, as well as classical information. An example of quantum information is the state of a qubit...
  
  s categorically.
• Special commutative dagger Frobenius algebra
  Frobenius algebra
  In mathematics, especially in the fields of representation theory and module theory, a Frobenius algebra is a finite dimensional unital associative algebra with a special kind of bilinear form which gives the algebras particularly nice duality theories. Frobenius algebras began to be studied in...
  
  s model the fact that certain processes yield classical information, that can be cloned or deleted, thus capturing classical communication
  Classical information channel
  In quantum information science, classical information channel is a communication channel that can be used to transmit classical information...
  
  .
• In early works, dagger biproduct
  Biproduct
  In category theory and its applications to mathematics, a biproduct of a finite collection of objects in a category with zero object is both a product and a coproduct. In a preadditive category the notions of product and coproduct coincide for finite collections of objects...
  
  s were used to study both classical communication
  Classical information channel
  In quantum information science, classical information channel is a communication channel that can be used to transmit classical information...
  
   and the superposition principle
  Superposition principle
  In physics and systems theory, the superposition principle , also known as superposition property, states that, for all linear systems, the net response at a given place and time caused by two or more stimuli is the sum of the responses which would have been caused by each stimulus individually...
  
  . Later, these two features have been separated.
• Complementary Frobenius algebra
  Frobenius algebra
  In mathematics, especially in the fields of representation theory and module theory, a Frobenius algebra is a finite dimensional unital associative algebra with a special kind of bilinear form which gives the algebras particularly nice duality theories. Frobenius algebras began to be studied in...
  
  s embody the principle of complementarity
  Complementarity (physics)
  In physics, complementarity is a basic principle of quantum theory proposed by Niels Bohr, closely identified with the Copenhagen interpretation, and refers to effects such as the wave–particle duality...
  
  , which is used to great effect in quantum computation.

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

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.
These diagrammatic languages trace back to Penrose's work in the early 1970s.

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.

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.
• Dagger commutative Frobenius algebras in the category of finite-dimensional Hilbert spaces correspond to orthogonal bases. This can be extended to arbitrary dimensions.
• Certain extra axioms guarantee that the scalars embed into the field of complex number
  Complex number
  A complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...
  
  s, namely the existence of finite dagger biproducts and dagger equalizers, well-pointedness, and a cardinality restriction on the scalars.
• 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.
• Special dagger commutative Frobenius algebras in the category of sets and relations correspond to discrete Abelian groupoids.
• Finding complementary basis structures in the category of sets and relations corresponds to solving combinatorical problems involving Latin squares.
• 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.

Categorical quantum mechanics as logic

Categorical quantum mechanics can also be seen as a type theoretic

Type theory

In mathematics, logic and computer science, type theory is any of several formal systems that can serve as alternatives to naive set theory, or the study of such formalisms in general...

 form of quantum logic

Quantum logic

In quantum mechanics, quantum logic is a set of rules for reasoning about propositions which takes the principles of quantum theory into account...

 that, in contrast to traditional quantum logic

Quantum logic

In quantum mechanics, quantum logic is a set of rules for reasoning about propositions which takes the principles of quantum theory into account...

, supports formal deductive reasoning. 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 and dagger complemented biproduct categories. 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

Spekkens Toy Model

The Spekkens toy model is a conceptually simple model, introduced by Robert Spekkens in 2004, to argue in favour of the epistemic view of quantum mechanics...

and enabled to pinpoint which structural ingredient causes it to be different from quantum theory.