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.[1] 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.
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]
Complementary Frobenius algebras embody the principle of
complementarity, which is used to great effect in quantum computation, as in the
ZX-calculus.[5]
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 Categories for quantum theory[10] and Picturing quantum processes.[11]
Diagrammatic calculus
One of the most notable features of categorical quantum mechanics is that the compositional structure can be faithfully captured by
string diagrams.[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]
Six axioms characterize the category of Hilbert spaces completely, fulfilling the reconstruction programme.[22] Two of these axioms concern a dagger and a tensor product, a third concerns biproducts.
Finding complementary basis structures in the category of sets and relations corresponds to solving combinatorical problems involving
Latin squares.[24]
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.[25]
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.[26] 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.[27][28] 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.[29]
Categorical quantum mechanics and DisCoCat
The
DisCoCat framework applies categorical quantum mechanics to
natural language processing.[30] 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.[31][32] 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.[33]
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