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Incategory theory, a branch ofmathematics,dagger compact categories (ordagger compact closed categories) first appeared in 1989 in the work ofSergio Doplicher and John E. Roberts on the reconstruction ofcompact topological groups from their category of finite-dimensional continuous unitary representations (that is,Tannakian categories).^[1] They also appeared in the work ofJohn Baez and James Dolan as an instance of semistrictk-tuplymonoidaln-categories, which describe generaltopological quantum field theories,^[2] forn = 1 andk = 3. They are a fundamental structure inSamson Abramsky andBob Coecke'scategorical quantum mechanics.^[3][4][5]

Dagger compact categories can be used to express and verify some fundamentalquantum information protocols, namely:teleportation,logic gate teleportation andentanglement swapping, and standard notions such as unitarity, inner-product, trace,Choi–Jamiolkowsky duality,complete positivity,Bell states and many other notions are captured by the language of dagger compact categories.^[3] All this follows from the completeness theorem, below.Categorical quantum mechanics takes dagger compact categories as a background structure relative to which other quantum mechanical notions like quantum observables and complementarity thereof can be abstractly defined. This forms the basis for a high-level approach toquantum information processing.

Adagger compact category is adagger symmetric monoidal category${\displaystyle \mathbf{C}}$ which is alsocompact closed, together with a relation to tie together the dagger structure to the compact structure. Specifically, the dagger is used to connect the unit to the counit, so that, for all${\displaystyle A}$ in${\displaystyle \mathbf{C}}$, the following diagram commutes:

To summarize all of these points:

• A category isclosed if it has aninternal hom functor; that is, if thehom-set of morphisms between two objects of the category is an object of the category itself (rather than ofSet).
• A category ismonoidal if it is equipped with an associativebifunctor${\displaystyle \mathbf{C}\otimes \mathbf{C}\to \mathbf{C}}$ that is associative,natural and has left and right identities obeying certaincoherence conditions.
• A monoidal category issymmetric monoidal, if, for every pairA,B of objects inC, there is an isomorphism${\displaystyle {\sigma }_{A,B}:A\otimes B\simeq B\otimes A}$ that isnatural in bothA andB, and, again, obeys certain coherence conditions (seesymmetric monoidal category for details).
• A monoidal category iscompact closed, if every object${\displaystyle A\in \mathbf{C}}$ has adual object${\displaystyle A^{\ast }}$. Categories with dual objects are equipped with two morphisms, theunit${\displaystyle {\eta }_A:I\to A^{\ast }\otimes A}$ and the counit${\displaystyle {\epsilon }_A:A\otimes A^{\ast }\to I}$, which satisfy certain coherence or yanking conditions.
• A category is adagger category if it is equipped with aninvolutivefunctor${\displaystyle †:{\mathbf{C}}^{op}\to \mathbf{C}}$ that is the identity on objects, but maps morphisms to their adjoints.
• A monoidal category isdagger symmetric if it is a dagger category and is symmetric, and has coherence conditions that make the various functors natural.

A dagger compact category is then a category that is each of the above, and, in addition, has a condition to relate the dagger structure to the compact structure. This is done by relating the unit to the counit via the dagger:

${\displaystyle {\sigma }_{A,A^{\ast }}\circ {\epsilon }_A^{†}={\eta }_A}$

shown in the commuting diagram above. In the categoryFdHilb of finite-dimensional Hilbert spaces, this last condition can be understood as defining the dagger (the Hermitian conjugate) as the transpose of the complex conjugate.

The following categories are dagger compact.

• The categoryFdHilb offinite dimensional Hilbert spaces andlinear maps. The morphisms arelinear operators between Hilbert spaces. The product is the usualtensor product, and the dagger here is theHermitian conjugate.
• The categoryRel ofSets and relations. The product is, of course, theCartesian product. The dagger here is just theopposite.
• The category offinitely generatedprojective modules over acommutative ring. The dagger here is just thematrix transpose.
• The categoryn-Cob ofcobordisms. Here, then-dimensional cobordisms are the morphisms, the disjoint union is the tensor, and the reversal of the objects (closed manifolds) is the dagger. Atopological quantum field theory can be defined as afunctor fromn-Cob intoFdHilb.^[6]
• The categorySpan(C) ofspans for any categoryC withfinite limits.

In all the examples listed except forn-Cob for sufficiently largen, every object is isomorphic to its dual: a compact dagger category with this property is called "self-dual". If${\displaystyle G}$ is a group andRep(${\displaystyle G}$) is its category of unitaryrepresentations on finite-dimensional Hilbert spaces, with the usual tensor product of representations, thenRep(${\displaystyle G}$) is always dagger compact, but it may or may not be self-dual, depending on the group${\displaystyle G}$.

Infinite-dimensional Hilbert spaces are not dagger compact, and are described bydagger symmetric monoidal categories.

Selinger showed that dagger compact categories admit a Joyal-Street style diagrammatic language^[7] and proved that dagger compact categories are complete with respect to finite dimensional Hilbert spaces^[8][9]i.e. an equational statement in the language of dagger compact categories holds if and only if it can be derived in the concrete category of finite dimensional Hilbert spaces and linear maps. There is no analogous completeness forRel orn-Cob.

This completeness result implies that various theorems from Hilbert spaces extend to this category. For example, theno-cloning theorem implies that there is no universal cloning morphism.^[10] Completeness also implies far more mundane features as well: dagger compact categories can be given a basis in the same way that a Hilbert space can have a basis. Operators can be decomposed in the basis; operators can have eigenvectors,etc.. This is reviewed in the next section.

The completeness theorem implies that basic notions from Hilbert spaces carry over to any dagger compact category. The typical language employed, however, changes. The notion of abasis is given in terms of acoalgebra. Given an objectA from a dagger compact category, a basis is acomonoid object${\displaystyle (A,\delta ,\epsilon )}$. The two operations are acopying orcomultiplication δ:A →A ⊗A morphism that is cocommutative and coassociative, and adeleting operation orcounit morphism ε:A →I . Together, these obey five axioms:^[11]

Comultiplicativity:

${\displaystyle (1_A\otimes \epsilon )\circ \delta =1_A=(\epsilon \otimes 1_A)\circ \delta }$

Coassociativity:

${\displaystyle (1_A\otimes \delta )\circ \delta =(\delta \otimes 1_A)\circ \delta }$

Cocommutativity:

${\displaystyle {\sigma }_{A,A}\circ \delta =\delta }$

Isometry:

${\displaystyle {\delta }^{†}\circ \delta =1_A}$

Frobenius law:

${\displaystyle ({\delta }^{†}\otimes 1_A)\circ (1_A\otimes \delta )=\delta \circ {\delta }^{†}}$

To see that these relations define a basis of a vector space in the traditional sense, write the comultiplication and counit usingbra–ket notation, and understanding that these are now linear operators acting on vectors${\displaystyle |j⟩}$ in a Hilbert spaceH:

and

The only vectors${\displaystyle |j⟩}$ that can satisfy the above five axioms must be orthogonal to one-another; the counit then uniquely specifies the basis. The suggestive namescopying anddeleting for the comultiplication and counit operators come from the idea that theno-cloning theorem andno-deleting theorem state that theonly vectors that it is possible to copy or delete are orthogonal basis vectors.

Given the above definition of a basis, a number of results for Hilbert spaces can be stated for compact dagger categories. We list some of these below, taken from^[11] unless otherwise noted.

• A basis can also be understood to correspond to anobservable, in that a given observable factors on (orthogonal) basis vectors. That is, an observable is represented by an objectA together with the two morphisms that define a basis:${\displaystyle (A,\delta ,\epsilon )}$.
• Aneigenstate of the observable${\displaystyle (A,\delta ,\epsilon )}$ is any object${\displaystyle \psi }$ for which

${\displaystyle \delta \circ \psi =\psi \otimes \psi }$

Eigenstates are orthogonal to one another.^{[clarification needed]}

• An object${\displaystyle \psi }$ iscomplementary to the observable${\displaystyle (A,\delta ,\epsilon )}$ if^{[clarification needed]}

${\displaystyle {\delta }^{†}\circ (\overline{\psi }\otimes \psi )={\epsilon }^{†}}$

(In quantum mechanics, a state vector${\displaystyle \psi }$ is said to be complementary to an observable if any measurement result is equiprobable. viz. an spin eigenstate ofS_x is equiprobable when measured in the basisS_z, or momentum eigenstates are equiprobable when measured in the position basis.)

• Two observables${\displaystyle (A,{\delta }_X,{\epsilon }_X)}$ and${\displaystyle (A,{\delta }_Z,{\epsilon }_Z)}$ are complementary if

${\displaystyle {\delta }_Z^{†}\circ {\delta }_X={\epsilon }_Z\circ {\epsilon }_X^{†}}$

• Complementary objects generateunitary transformations. That is,

${\displaystyle {\delta }^{†}\circ (\psi \otimes 1_A)}$

is unitary if and only if${\displaystyle \psi }$ is complementary to the observable${\displaystyle (A,\delta ,\epsilon )}$

• Dagger-compact category at thenLab

• Monoidal categories
• Dagger categories

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