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| In [[mathematics]], '''dagger compact categories''' (or '''dagger compact closed categories''') first appeared in 1989 in the work of Doplicher and Roberts on the reconstruction of [[compact topological group]]s from their category of finite-dimensional continuous unitary representations (that is, [[Tannakian category|Tannakian categories]]).<ref name=AQFT>S. Doplicher and J. Roberts, A new duality theory for compact groups, Invent. Math. 98 (1989) 157-218.</ref> They also appeared in the work of [[John Baez|Baez]] and Dolan as an instance of semistrict k-tuply monoidal [[n-category|n-categories]], which describe general [[topological quantum field theories]],<ref name=HDATQFT>J. C. Baez and J. Dolan, ''[http://arxiv.org/abs/q-alg/9503002 Higher-dimensional Algebra and Topological Quantum Field Theory]'', J.Math.Phys. 36 (1995) 6073-6105</ref> for n = 1 and k = 3. They are a fundamental structure in [[Samson Abramsky|Abramsky]] and [[Bob Coecke|Coecke]]'s [[categorical quantum mechanics]],<ref name=QProtocols>[[Samson Abramsky]] and [[Bob Coecke]], ''[http://arxiv.org/abs/quant-ph/0402130 A categorical semantics of quantum protocols]'', Proceedings of the 19th IEEE conference on Logic in Computer Science (LiCS'04). IEEE Computer Science Press (2004).</ref><ref name=CQM>S. Abramsky and B. Coecke, ''[http://arxiv.org/abs/0808.1023 Categorical quantum mechanics]". In: Handbook of Quantum Logic and Quantum Structures, K. Engesser, D. M. Gabbay and D. Lehmann (eds), pages 261–323. Elsevier (2009).</ref><ref>Abramsky and Coecke used the term strongly compact closed categories, since a dagger compact category is a [[compact closed category]] augmented with a covariant involutive monoidal endofunctor.</ref>
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| ==Overview==
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| Dagger compact categories can be used to express and verify some fundamental [[quantum computing|quantum information]] protocols, namely: [[quantum teleportation|teleportation]], [[logic gate teleportation]] and [[quantum teleportation|entanglement swapping]], and standard notions such as unitarity, inner-product, trace, [[channel-state duality|Choi-Jamiolkowsky duality]], [[completely positive|complete positivity]], [[Bell state]]s and many other notions are captured by the language of dagger compact categories.<ref name="QProtocols"/> 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 to [[quantum computing|quantum information]] processing.
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| ==Formal definition==
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| A '''dagger compact category''' is a [[dagger symmetric monoidal category]] <math>\mathbf{C}</math> which is also [[compact closed category|compact 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 <math> A </math> in <math> \mathbf{C}</math>, the following diagram commutes:
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| [[File:Dagger compact category (diagram).png|center|240px]]
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| To summarize all of these points:
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| * A category is [[closed category|closed]] if it has an [[internal hom functor]]; that is, if the [[hom-set]] of morphisms between two objects of the category is an object of the category itself (rather than of '''Set''').
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| * A category is [[monoidal category|monoidal]] if it is equipped with an associative [[bifunctor]] <math>\mathbf{C} \otimes \mathbf{C} \to \mathbf{C}</math> that is associative, [[natural transformation|natural]] and has left and right identities obeying certain [[coherence condition]]s.
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| * A monoidal category is [[symmetric monoidal category|symmetric monoidal]], if, for every pair ''A'', ''B'' of objects in ''C'', there is an isomorphism <math>\sigma_{AB}: A \otimes B \simeq B \otimes A</math> that is [[natural transformation|natural]] in both ''A'' and ''B'', and, again, obeys certain coherence conditions (see [[symmetric monoidal category]] for details).
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| * A monoidal category is [[compact closed category|compact closed]], if every object <math>A \in \mathbf{C}</math> has a [[dual object]] <math>A^*</math>. Categories with dual objects are equipped with two morphisms, the [[unit (category theory)|unit]] <math>\eta_A:I\to A^*\otimes A</math> and the counit <math>\varepsilon_A:A\otimes A^*\to I</math>, which satisfy certain coherence or [[yanking condition]]s.
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| * A category is a [[dagger category]] if it is equipped with an [[Involution (mathematics)|involutive]] [[functor]] <math>\dagger\colon \mathbf{C}^{op}\rightarrow\mathbf{C}</math> that is the identiy on objects, but maps morphisms to their adjoints.
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| * A monoidal category is [[dagger symmetric monoidal category|dagger symmetric]] if it is a dagger category and is symmetric, and has coherence conditions that make the various functors natural.
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| 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:
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| :<math>\sigma_{A\otimes A^*} \circ\varepsilon^\dagger_A = \eta_A</math>
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| shown in the commuting diagram above. In the category '''FdHilb''' of finite-dimensional Hilbert spaces, this last condition can be understood as defining the dagger (the Hermitian conjugagte) as the transpose of the complex conjugate.
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| ==Examples==
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| The following categories are dagger compact.
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| * The category '''FdHilb''' of [[Category of finite dimensional Hilbert spaces|finite dimensional Hilbert spaces]] and [[linear maps]]. The morphisms are [[linear operator]]s between Hilbert spaces. The product is the usuall [[tensor product]], and the dagger here is the [[Hermitian conjugate]].
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| * The category '''Rel''' of [[Category of relations|Sets and relations]]. The product is, of course, the [[Cartesian product]]. The dagger here is just the [[opposite (category theory)|opposite]].
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| * The category of [[Finitely generated module|finitely generated]] [[projective modules]] over a [[commutative ring]]. The dagger here is just the [[matrix transpose]].
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| * The category '''nCob''' of [[cobordism]]s. Here, the n-dimensional cobordisms are the morphisms, the disjoint union is the tensor, and the reversal of the objects (closed manifolds) is the dagger. A [[topological quantum field theory]] can be defined as a [[functor]] from '''nCob''' into '''FdHilb'''.<ref>M. Atiyah, "Topological quantum field theories". ''Inst. Hautes Etudes Sci. Publ. Math.'' '''68''' (1989), pp. 175–186.</ref>
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| * The category '''Span'''(''C'') of [[span (category theory)|spans]] for any category ''C'' with [[limit (category theory)|finite limits]].
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| Infinite-dimensional Hilbert spaces are not dagger compact, and are described by [[dagger symmetric monoidal category|dagger symmetric monoidal categories]].
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| ==Structural theorems==
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| Selinger showed that dagger compact categories admit a Joyal-Street style diagrammatic language<ref>P. Selinger, ''[http://www.mscs.dal.ca/~selinger/papers.html#dagger Dagger compact closed categories and completely positive maps]'', Proceedings of the 3rd International Workshop on Quantum Programming Languages, Chicago, June 30 - July 1 (2005).</ref> and proved that dagger compact categories are complete with respect to finite dimensional Hilbert spaces<ref>P. Selinger, ''[http://www.mscs.dal.ca/~selinger/papers.html#finhilb Finite dimensional Hilbert spaces are complete for dagger compact closed categories]'', Proceedings of the 5th International Workshop on Quantum Programming Languages, Reykjavik (2008).</ref><ref>M. Hasegawa, M. Hofmann and G. Plotkin, "Finite dimensional vector spaces are complete for traced symmetric monoidal categories", LNCS '''4800''', (2008), pp. 367–385.</ref> ''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 for '''Rel''' or '''nCob''' (obviously, for if there were, they'd be Hilbert spaces!)
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| This completeness result implies that various theorems from Hilbert spaces extend to this category. For example, the [[no-cloning theorem]] implies that there is no universal cloning morphism.<ref>S. Abramsky, "No-Cloning in categorical quantum mechanics", (2008) ''Semantic Techniques for Quantum Computation'', I. Mackie and S. Gay (eds), Cambridge University Press</ref> 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.
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| == Basis==
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| 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 a [[basis (linear algebra)|basis]] is given in terms of a [[coalgebra]]. Given an object ''A'' from a dagger compact catgeory, a basis is a [[comonoid object]] <math>(A,\delta,\varepsilon)</math>. The two operations are ''copying'' or [[comultiplication]] δ: ''A'' → ''A'' ⊗ ''A'' that is cocommutative and coassociative, and a ''deleting'' operation or [[counit]] and ε: ''A'' → ''I'' . Together, these obey five axioms:<ref name="coecke"/>
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| Comultiplicativity:
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| :<math>(1_A \otimes \varepsilon) \circ \delta =1_A = (\varepsilon \otimes 1_A) \circ \delta</math>
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| Coassociativity:
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| :<math>(1_A \otimes \delta) \circ \delta = (\delta \otimes 1_A) \circ \delta</math>
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| Cocommutativity:
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| :<math>\sigma_{A,A} \circ \delta = \delta</math>
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| Isometry:
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| :<math>\delta^\dagger \circ \delta = 1_A</math>
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| [[Frobenius algebra|Frobenius law]]: <!-- this is a horrible link, need something else -->
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| :<math>(\delta^\dagger \otimes 1_A) \circ (1_A \otimes \delta) = \delta \circ \delta^\dagger</math>
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| To see that these relations define a basis of a vector space in the traditional sense, write the comultiplication and counit using [[bra-ket notation]], and understanding that these are now linear operators acting on vectors | ''j'' > in a Hilbert space ''H'':
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| :<math>\begin{align}
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| \delta : H &\to H\otimes H \\
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| |j\rangle & \mapsto |j\rangle\otimes |j\rangle = |j j \rangle \\
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| \end{align}</math>
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| and
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| :<math>\begin{align}
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| \varepsilon : H &\to \mathbb{C} \\
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| |j\rangle & \mapsto 1\\
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| \end{align}</math>
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| The only vectors | ''j'' > that can satisfy the above five axioms must be orthogonal to one-another; the counit then uniquely specifies the basis. The suggestive names ''copying'' and ''deleting'' for the comultiplication and counit operators come from the idea that the [[no-cloning theorem]] and [[no-deleting theorem]] state that the ''only'' vectors that it is possible to copy or delete are orthogonal basis vectors.
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| == General results==
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| 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<ref name="coecke">Bob Coecke, "Quantum Picturalism", (2009) ''Contemporary Physics'' vol '''51''', pp59-83. ([http://arxiv.org/abs/0908.1787 ArXiv 0908.1787])</ref> unless otherwise noted.
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| * A basis can also be understood to correspond to an [[observable]], in that a given observable factors on (orthogonal) basis vectors. That is, an observable is denoted as and object ''A'' and the two functors that define the basis: <math>(A, \delta_A, \varepsilon_A)</math>.
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| * An [[eigenstate]] of a dagger compact category is any object <math>\psi</math> for which
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| ::<math>\delta \circ \psi = \psi \otimes \psi</math>
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| :Eigensates are orthogonal to one another.
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| * An object <math>\psi</math> is [[Complementarity (physics)|complementary]] to the observable <math>(A, \delta_A, \varepsilon_A)</math> if
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| ::<math>\delta^\dagger \circ (\overline\psi \otimes \psi) = \varepsilon^\dagger</math>
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| :(In quantum mechanics, a state vector <math>\psi</math> is said to be complementary to an observable if any measurement result is equiprobable. viz. an spin eigenstate of ''S''<sub>x</sub> is equiprobable when measured in the basis ''S''<sub>z</sub>).
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| * Two observables <math>(A, \delta_X, \varepsilon_X)</math> and <math>(A, \delta_Z, \varepsilon_Z)</math> are complementary if
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| ::<math>\delta^\dagger_Z \circ \delta_X = \varepsilon_Z \circ \varepsilon_X^\dagger</math>
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| * Complementary objects generate [[unitary transformation]]s. That is,
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| ::<math>\delta^\dagger \circ (\psi\otimes 1_A)</math>
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| :is unitary if and only if <math>\psi</math> is complementary to the observable <math>(A, \delta, \varepsilon)</math>
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| ==References==
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| {{reflist}}
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| * {{nlab|id=dagger-compact+category|title=Dagger-compact category}}
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| [[Category:Monoidal categories]]
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| [[Category:Dagger categories]]
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