Dagger compact category

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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 groups from their category of finite-dimensional continuous unitary representations (that is, Tannakian categories).[1] They also appeared in the work of Baez and Dolan as an instance of semistrict k-tuply monoidal n-categories, which describe general topological quantum field theories,[2] for n = 1 and k = 3. They are a fundamental structure in Abramsky and Coecke's categorical quantum mechanics.[3][4][5]

Overview[edit]

Dagger compact categories can be used to express and verify some fundamental quantum information protocols, namely: teleportation, logic gate teleportation and entanglement 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 to quantum information processing.

Formal definition[edit]

A dagger compact category is a dagger symmetric monoidal category \mathbf{C} which is also 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  A in  \mathbf{C}, the following diagram commutes:

Dagger compact category (diagram).png

To summarize all of these points:

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:

\sigma_{A\otimes A^*} \circ\varepsilon^\dagger_A = \eta_A

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 conjugate) as the transpose of the complex conjugate.

Examples[edit]

The following categories are dagger compact.

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

Structural theorems[edit]

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 for Rel or nCob (obviously, for if there were, they'd be Hilbert spaces!)

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.[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.

Basis[edit]

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 is given in terms of a coalgebra. Given an object A from a dagger compact category, a basis is a comonoid object (A,\delta,\varepsilon). The two operations are copying or comultiplication δ: AAA that is cocommutative and coassociative, and a deleting operation or counit and ε: AI . Together, these obey five axioms:[11]

Comultiplicativity:

(1_A \otimes \varepsilon) \circ \delta =1_A = (\varepsilon \otimes 1_A) \circ \delta

Coassociativity:

(1_A \otimes \delta) \circ \delta = (\delta \otimes 1_A) \circ \delta

Cocommutativity:

\sigma_{A,A} \circ \delta = \delta

Isometry:

\delta^\dagger \circ \delta = 1_A

Frobenius law:

(\delta^\dagger \otimes 1_A) \circ (1_A \otimes \delta) = \delta \circ \delta^\dagger

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:

\begin{align}
\delta : H &\to H\otimes H \\
|j\rangle & \mapsto |j\rangle\otimes |j\rangle = |j j \rangle \\
\end{align}

and

\begin{align}
\varepsilon : H &\to \mathbb{C} \\
|j\rangle & \mapsto 1\\
\end{align}

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.

General results[edit]

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 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: (A, \delta_A, \varepsilon_A).
  • An eigenstate of a dagger compact category is any object \psi for which
\delta \circ \psi = \psi \otimes \psi
Eigensates are orthogonal to one another.
\delta^\dagger \circ (\overline\psi \otimes \psi) = \varepsilon^\dagger
(In quantum mechanics, a state vector \psi is said to be complementary to an observable if any measurement result is equiprobable. viz. an spin eigenstate of Sx is equiprobable when measured in the basis Sz).
  • Two observables (A, \delta_X, \varepsilon_X) and (A, \delta_Z, \varepsilon_Z) are complementary if
\delta^\dagger_Z \circ \delta_X = \varepsilon_Z \circ \varepsilon_X^\dagger
\delta^\dagger \circ (\psi\otimes 1_A)
is unitary if and only if \psi is complementary to the observable (A, \delta, \varepsilon)

References[edit]

  1. ^ S. Doplicher and J. Roberts, A new duality theory for compact groups, Invent. Math. 98 (1989) 157-218.
  2. ^ J. C. Baez and J. Dolan, Higher-dimensional Algebra and Topological Quantum Field Theory, J.Math.Phys. 36 (1995) 6073-6105
  3. ^ a b Samson Abramsky and Bob Coecke, A categorical semantics of quantum protocols, Proceedings of the 19th IEEE conference on Logic in Computer Science (LiCS'04). IEEE Computer Science Press (2004).
  4. ^ S. Abramsky and B. Coecke, 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).
  5. ^ 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.
  6. ^ M. Atiyah, "Topological quantum field theories". Inst. Hautes Etudes Sci. Publ. Math. 68 (1989), pp. 175–186.
  7. ^ P. Selinger, Dagger compact closed categories and completely positive maps, Proceedings of the 3rd International Workshop on Quantum Programming Languages, Chicago, June 30 - July 1 (2005).
  8. ^ P. Selinger, Finite dimensional Hilbert spaces are complete for dagger compact closed categories, Proceedings of the 5th International Workshop on Quantum Programming Languages, Reykjavik (2008).
  9. ^ M. Hasegawa, M. Hofmann and G. Plotkin, "Finite dimensional vector spaces are complete for traced symmetric monoidal categories", LNCS 4800, (2008), pp. 367–385.
  10. ^ S. Abramsky, "No-Cloning in categorical quantum mechanics", (2008) Semantic Techniques for Quantum Computation, I. Mackie and S. Gay (eds), Cambridge University Press
  11. ^ a b Bob Coecke, "Quantum Picturalism", (2009) Contemporary Physics vol 51, pp59-83. (ArXiv 0908.1787)