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Concept
Categorical quantum mechanics
Summary
Categorical quantum mechanics is the study of quantum foundations and quantum information using paradigms from mathematics and computer science, notably . 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. Categorical quantum mechanics is entry 18M40 in MSC2020.
Mathematically, the basic setup is captured by a : 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 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. 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.
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.
In early works, dagger biproducts were used to study both classical communication and the superposition principle. Later, these two features have been separated.
Complementary Frobenius algebras embody the principle of complementarity, which is used to great effect in quantum computation, as in the ZX-calculus.
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